Plant control system

ABSTRACT

A plant control system has a reference value setting unit for variably setting a reference value for an air-fuel ratio to be given to an exhaust system including a catalytic converter, depending on a component based on an adaptive control law of a manipulated variable of the air-fuel ratio generated by a controller according to an adaptive sliding mode control process in order to converge an output of an O 2  sensor disposed downstream of the catalytic converter to a target value. The plant control system also has an estimator for estimating the difference between an output of the O 2  sensor after the dead time of the exhaust system and a target value therefor, using the difference between the set reference value and a detected value of the air-fuel ratio, and giving the estimated difference to the controller.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a plant control system.

2. Description of the Prior Art

The applicant of the present application has proposed an air-fuel ratiocontrol system having an exhaust gas sensor for detecting theconcentration of a certain component of an exhaust gas that has passedthrough a catalytic converter such as a three-way catalytic converterdisposed in the exhaust passage of an internal combustion engine, suchas an O₂ sensor for detecting the concentration of oxygen in the exhaustgas, the exhaust gas sensor being disposed downstream of the catalyticconverter. The system controls the air-fuel ratio of the internalcombustion engine, more accurately, the air-fuel ratio of an air-fuelmixture to be combusted by the internal combustion engine, in order toconverge an output of the O₂ sensor, i.e., the detected value of theoxygen concentration, to a predetermined target value for enabling thecatalytic converter to have a desired purifying ability irrespective ofthe aging of the catalytic converter. See, for example, U.S. patentapplication Ser. No. 09/311353, now U.S. Pat. No. 6,188,953 U.S. Pat.No. 6,112,517 and Japanese patent application No. 11-93740 (U.S. Pat.No. 6,079,205).

According to the disclosed technology, a manipulated variable formanipulating the air-fuel ratio of the internal combustion engine,specifically, a target air-fuel ratio or a quantity defining such atarget air-fuel ratio, is successively generated in given control cyclesin order to converge the output of the O₂ sensor to its target valuebased on a feedback control process. An exhaust gas sensor (hereinafterreferred to as an “air-fuel ratio sensor) for detecting the air-fuelratio of the exhaust gas that enters the catalytic converter,specifically, the air-fuel ratio of the air-fuel mixture that has beenburned by the internal combustion engine, is disposed upstream of thecatalytic converter. The amount of fuel supplied to the internalcombustion engine is regulated so as to converge the output of theair-fuel ratio sensor, i.e., the detected value of the air-fuel ratio,to a target air-fuel ratio defined by the manipulated variable forthereby controlling the air-fuel ratio of the internal combustion engineat the target air-fuel ratio.

Such air-fuel ratio control for the internal combustion engine iscapable of converging the output of the O₂ sensor disposed downstream ofthe catalytic converter to its target value for thereby enabling thecatalytic converter to have a desired purifying ability.

In the above proposed air-fuel ratio control system, the feedbackcontrol process using the output of the air-fuel ratio sensor disposedupstream of the catalytic converter is carried out for controlling theair-fuel ratio of the internal combustion engine at the target air-fuelratio. However, it is also possible to control the air-fuel ratio of theinternal combustion engine at the target air-fuel ratio according to afeed-forward control process by determining the amount of fuel suppliedto the internal combustion engine from the target air-fuel ratio using amap or the like.

In the above air-fuel ratio control system, the O₂ sensor is used as theexhaust gas sensor disposed downstream of the catalytic converter.However, the exhaust gas sensor may comprise an NOx sensor, a CO sensor,an HC sensor, or another exhaust gas sensor. It is possible to enablethe catalytic converter to have a desired purifying ability bycontrolling the air-fuel ratio of the internal combustion engine so asto converge the output of such an exhaust gas sensor to a suitabletarget value.

In the above conventional air-fuel ratio control system, the exhaustsystem, including the catalytic converter, which ranges from a positionupstream of the catalytic converter to the O₂ sensor downstream of thecatalytic converter may be considered to be a plant for generating theoutput of the O₂ sensor from the air-fuel ratio of the internalcombustion engine (the air-fuel ratio as detected by the air-fuel ratiosensor). The internal combustion engine may be considered to be anactuator for generating an exhaust gas having an air-fuel ratio to besupplied to the plant. Thus, the air-fuel ratio control system may beexpressed as a system for generating a manipulated variable to controlthe input (air-fuel ratio) to the plant (=an output from the actuator)to converge the output of the O₂ sensor as the output of the plant to agiven target value, and controlling operation of the internal combustionengine as the actuator based on the manipulated variable.

The catalytic converter generally has a relatively long dead time whichtends to adversely affect the control process of converting the outputof the O₂ sensor to the target value. In the air-fuel ratio controlsystem, the behavior of the exhaust system including the catalyticconverter and ranging from the position upstream of the catalyticconverter to the O₂ sensor downstream of the catalytic converter ismodeled. The output of the O₂ sensor after the dead time of the exhaustsystem is successively estimated according to an algorithm constructedon the basis of the model of the exhaust system. The manipulatedvariable is generated using the estimated value of the output of the O₂sensor (specifically, the manipulated variable is generated in order toconverge the estimated value of the output of the O₂ sensor to thetarget value) for thereby compensating for the effect of the dead time.

In modeling the exhaust system including the catalytic converter, theexhaust system is regarded as a system for generating the differencebetween the output of the O₂ sensor and its target value with a responsedelay and the dead time, from the difference (hereinafter referred to asa “differential air-fuel ratio”) between the air-fuel ratio of theair-fuel mixture combusted by the internal combustion engine, i.e., theair-fuel ratio detected by the air-fuel ratio sensor, and apredetermined constant reference value with respect to the air-fuelratio. The algorithm for estimating the output of the O₂ sensor afterthe dead time of the exhaust system is constructed on the basis of themodel of the exhaust system.

With the input (differential air-fuel ratio) to the exhaust system thatis modeled and the output thereof (the difference between the output ofthe O₂ sensor and its target value) being expressed as differences, thealgorithm for estimating the output of the O₂ sensor after the dead timeof the exhaust system can be simplified.

The applicant has also proposed a technique for compensating for notonly the dead time of the exhaust system, but also the effect of deadtimes of the internal combustion engine and a controller (hereinafterreferred to as an “engine controller”) which controls operation of theinternal combustion engine based on the manipulated variable, in U.S.patent application Ser. No. 09/311353. This process is proposed becausethe dead times of the internal combustion engine and the enginecontroller may adversely affect the control process of converting theoutput of the O₂ sensor to the target value, depending on an operatingstate of the internal combustion engine (more specifically, the state ofthe rotational speed thereof) for which the control process ofconverting the output of the O₂ sensor to the target value is carriedout.

According to the above proposed technique, the exhaust system is modeledas described above, and a system comprising the internal combustionengine and the engine controller is modeled as a system for generatingan air-fuel ratio to be given to the exhaust system, with only the deadtime of the air-fuel ratio control system, from the manipulatedvariable, i.e., a system in which the air-fuel ratio to be given to theexhaust system is uniquely defined by the manipulated variable prior tothe dead time. According to an algorithm constructed on the basis ofthese models, the output of the O₂ sensor after a total dead timerepresenting the sum of the dead time of the exhaust system and the deadtime of the air-fuel ratio control system is successively estimated, andthe manipulated variable is generated using the estimated value.Therefore, it is possible to converge the output of the O₂ sensor to thetarget value while compensating for not only the dead time of theexhaust system, but also the dead time of the air-fuel ratio controlsystem.

In estimating the output of the O₂ sensor after the total dead time,with the input (differential air-fuel ratio) to the exhaust system andthe output thereof (the difference between the output of the O₂ sensorand its target value) being expressed as differences, the algorithm forestimating the output of the O₂ sensor can be simplified.

According to the above technique, a parameter for defining the behaviorof the model of the exhaust system is successively identified using theoutput data of the air-fuel ratio sensor and the O₂ sensor. Thealgorithm for estimating the output of the O₂ sensor after the dead timeof the exhaust system or the output of the O₂ sensor after the totaldead time which is the sum of the dead time of the exhaust system andthe dead time of the air-fuel ratio control system uses the identifiedparameter of the model of the exhaust system. Furthermore, the abovetechnique uses a sliding mode control process constructed on the basisof the model of the exhaust system as the feedback control process forgenerating the manipulated variable.

It has been found as a result of a further study by the inventors of thepresent application that when the output of the air-fuel ratio sensorsuffers a steady offset from a normal output due to deterioration of theair-fuel ration sensor or when the actual air-fuel ratio is subject to asteady error with respect to the target air-fuel ratio due to anaging-induced characteristic change of the internal combustion engine,the estimated value of the output of the O₂ sensor after the dead timeof the exhaust system or the estimated value of the output of the O₂sensor after the total dead time which is the sum of the dead time ofthe exhaust system and the dead time of the air-fuel ratio controlsystem tends to suffer a steady error with respect to a true value. Ifthe estimated value of the output of the O₂ sensor suffers such anerror, then when the air-fuel ratio of the internal combustion engine iscontrolled on the basis of the manipulated variable generated using theestimated value, the accuracy with which to converge the output of theO₂ sensor to the target value is lowered.

In addition, when the output of the air-fuel ratio sensor suffers anoffset, the identified value of the parameter of the exhaust systemmodel used for estimating the output of the O₂ sensor is likely tosuffer an error. Because of such an error, the estimated value of theoutput of the O₂ sensor may be subject to an error.

Moreover, with the above process of generating the manipulated variableaccording to the feedback control process constructed on the basis ofthe model of the exhaust system, when the output of the air-fuel ratiosensor suffers an offset, the quick response of the control process forconverging the output of the O₂ sensor to the target value is reduced.

It has been desired to eliminate the above drawbacks.

In another application, a manipulated variable for controlling an inputto an arbitrary plant is generated to converge a detected value of anoutput of the plant to a predetermined target value, and the operationof an actuator for generating the input to the plant is controlled onthe basis of the manipulated variable. The above drawbacks are alsocaused in such an application when the output of an O₂ sensor isestimated on the basis of a model of the plant which is constructed inthe same manner as with the above technique in order to compensate forthe dead time of the plant and the dead times of the actuator and itscontroller, and the manipulated variable is generated using theestimated value.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide a plantcontrol system for controlling an input to a plant in order to convergean output of a detecting means for detecting an output of the plant to apredetermined target value, the plant control system being capable ofincreasing the accuracy and quick response of a control process ofconverging the output of the detecting means to the target value whilecompensating for the effect of a dead time of the plant.

Another object of the present invention is to provide a plant controlsystem for controlling a plant which comprises an exhaust system rangingfrom a position upstream of a catalytic converter disposed in an exhaustpassage of an internal combustion engine for purifying an exhaust gasfrom the internal combustion engine, to a position downstream of thecatalytic converter, by controlling the air-fuel ratio of the internalcombustion engine as an input to the exhaust system in order to convergean output of an exhaust gas sensor for detecting the concentration of acertain component of the exhaust gas downstream of the catalyticconverter, to a predetermined target value.

To achieve the above objects, there is provided in accordance with afirst aspect of the present invention a plant control system forcontrolling a plant, comprising an actuator for generating an input tothe plant, first detecting means for detecting an output from the plant,manipulated variable determining means for sequentially generating amanipulated variable which manipulates the input to the plant toconverge the output from the first detecting means to a predeterminedtarget value, actuator control means for controlling operation of theactuator based on the manipulated variable to manipulate the input tothe plant, estimating means for sequentially generating datarepresenting an estimated value of the output from the first detectingmeans after a dead time of the plant according to a predeterminedalgorithm constructed based on a model of a behavior of the plant as asystem for generating the difference between the output from the firstdetecting means and the target value with a response delay and the deadtime from the difference between the input to the plant and apredetermined reference value, the manipulated variable determiningmeans comprising means for generating the manipulated variable using thedata generated by the estimating means, and reference value variablesetting means for variably setting the predetermined reference valuedepending on the manipulated variable generated by the manipulatedvariable determining means.

According to the first aspect of the present invention, datarepresenting an estimated value of the output (a detected value of theoutput of the plant) from the first detecting means after a dead time ofthe plant is sequentially generated according to a predeterminedalgorithm constructed based on a model of the plant. The dead time ofthe plant is specifically a time required until the input at each pointof time to the plant is reflected in the output from the first detectingmeans. In the first aspect of the present invention, using the datarepresenting the estimated value of the output from the first detectingmeans after the dead time, a manipulated variable which manipulates theinput to the plant to converge the output from the first detecting meansto a predetermined target value is sequentially generated, and operationof the actuator is controlled based on the manipulated variable tomanipulate the output of the actuator which is the input to the plant.

According to the finding of the inventors of the present application, ifthe reference value relative to the difference (which may be referred toas a plant differential input in the description of the invention)between the input to the plant expressed by the model and thepredetermined reference value is constant as is conventional, then whenthe input to the plant (=the output of the actuator) manipulated basedon the manipulated variable contains steady disturbance, the estimatedvalue after the dead time of the output from the first detecting meansrepresented by the data generated by the estimating means tends tosuffer a steady error with respect to a true value. Hence, when themanipulated variable is generated using the estimated value tomanipulate the input to the plant, the output from the first detectingmeans is also liable to suffer a steady error with respect to the targetvalue. In such a situation, when the reference value is appropriatelychanged depending on the manipulated variable generated by themanipulated variable generating means, the accuracy of the estimatedvalue of the output from the first detecting means can be increased.

According to the present invention, the predetermined reference value isvariably set depending on the manipulated variable generated by themanipulated variable determining means. In this manner, the accuracy ofthe estimated value of the output from the first detecting means afterthe dead time of the plant can be increased. It is thus possible toincrease the accuracy of the control process for converging the outputfrom the first detecting means to the target value while compensatingfor the effect of the dead time of the plant.

The manipulated variable generated by the manipulated variablegenerating means may comprise a target input to the plant, a targetvalue for the plant differential input, or a corrective variable for theamount of operation of the actuator. Using the data representing theestimated value of the output from the first detecting means, themanipulated variable may be generated by a feedback control process forconverging the estimated value of the output from the first detectingmeans, which is represented by the data, to the target value and hencefor converging the output from the first detecting means to the targetvalue.

In the first aspect of the present invention, the plant comprises anexhaust system ranging from a position upstream of a catalytic converterdisposed in an exhaust passage of an internal combustion engine forpurifying an exhaust gas from the internal combustion engine, to aposition downstream of the catalytic converter, the exhaust systemincluding the catalytic converter, the input to the plant comprising anair-fuel ratio of an air-fuel mixture combusted in the internalcombustion engine as the actuator for generating the exhaust gas whichenters the catalytic converter, the output from the plant comprising theconcentration of a component of the exhaust gas having passed throughthe catalytic converter.

Be cause the accuracy of the estimated value of the output (a detectedvalue of the concentration of the component) from the first detectingmeans represented by the data generated by the estimating means isincreased, the accuracy of the control process for converging the outputfrom the first detecting means to the target value can be increasedwhile compensating for the effect of the dead time of the exhaustsystem, i.e., a time required until the air-fuel ratio at each point oftime given to the exhaust system is reflected in the output from thefirst detecting means. As a result, the purifying capability of thecatalytic converter in the exhaust system is reliably achieved.

In this case, the manipulated variable generated by the manipulatedvariable generating means may comprise a target air-fuel ratio for theair-fuel mixture combusted in the internal combustion engine, a targetvalue for the difference (the plant differential input) between theair-fuel ratio of the air-fuel mixture and the reference value, or acorrective variable for the amount of fuel supplied to the internalcombustion engine. The actuator control means adjusts the amount of fuelsupplied to the internal combustion engine depending on the manipulatedvariable to manipulate the air-fuel ratio as the input given to theexhaust system (plant).

In the first aspect, more specifically, the plant control system furthercomprises second detecting means for detecting the input to the plant,the algorithm comprising an algorithm for generating the datarepresenting the estimated value of the output from the first detectingmeans after the dead time, using data representing the differencebetween the output of the first detecting means and the target value,data representing the difference (corresponding to a detected value ofthe plant differential input) between an output from the seconddetecting means and the reference value, and parameters of the model ofthe plant which define the behavior of the model of the plant.

When the input to the plant is detected by the second detecting means,the data representing the estimated value of the output from the firstdetecting means after the dead time, specifically, the estimated valueof the output of the first detecting means or the estimated value of theplant differential output, can be generated by using data representingthe difference (which may be referred to as a plant differential output)between the output of the first detecting means and the target value,data representing a detected value of the plant differential input, andparameters of the model of the plant which define the behavior of themodel of the plant.

The parameters comprise parameters to be set to certain values indefining the behavior of the model. When the manipulated variable is atarget input to the plant or a target value for the plant differentialinput, if an error between the target input to the plant determineddepending on the manipulated variable and an actual input issufficiently small, then it is possible to use the manipulated variableinstead of the output from the second detecting means which correspondsto the plant differential input, in order to generate the datarepresenting the estimated value of the output from the first detectingmeans.

To achieve the above objects, there is provided in accordance with asecond aspect of the present invention a plant control system forcontrolling a plant, comprising an actuator for generating an input tothe plant, first detecting means for detecting an output from the plant,manipulated variable determining means for sequentially generating amanipulated variable which manipulates the input to the plant toconverge the output from the first detecting means to a predeterminedtarget value, actuator control means for controlling operation of theactuator based on the manipulated variable to manipulate the input tothe plant, estimating means for sequentially generating datarepresenting an estimated value of the output from the first detectingmeans after a total dead time which is the sum of a first dead time ofthe plant and a second dead time of an input manipulating system,according to a predetermined algorithm constructed based on a model of abehavior of the plant as a system for generating the difference betweenthe output from the first detecting means and the target value with aresponse delay and the first dead time from the difference between theinput to the plant and a predetermined reference value, and a model of abehavior of the input manipulating system as a system comprising theactuator control means and the actuator for generating the input to theplant with the second dead time from the manipulated variable, themanipulated variable determining means comprising means for generatingthe manipulated variable using the data generated by the estimatingmeans, and reference value variably setting means for variably settingthe predetermined reference value depending on the manipulated variablegenerated by the manipulated variable determining means.

According to the second aspect of the present invention, datarepresenting an estimated value of the output from the first detectingmeans after a total dead time which is the sum of a first dead time ofthe plant and a second dead time of an input manipulating system issequentially generated according to a predetermined algorithmconstructed based on a model of the plant and a model of the inputmanipulating system (which comprises the actuator and the actuatorcontrol means). The first dead time of the plant is a time requireduntil the input at each point of time to the plant is reflected in theoutput from the first detecting means, and the second dead time of theinput manipulating system is a time required until the manipulatedvariable at each point of time given to the input manipulating system isreflected in the actual output from the actuator which is given to theplant.

In the second aspect, using the data representing the estimated value ofthe output from the first detecting means after the total dead timewhich is the sum of the first dead time and the second dead time, themanipulated variable for manipulating the input to the plant in order toconverge the output from the first detecting means to the target valueis sequentially generated, and operation of the actuator is controlledbased on the manipulated variable to manipulate the output of theactuator which is the input to the plant.

As with the first aspect of the present invention, if the referencevalue relative to the plant differential input is constant as isconventional, then when the input to the plant (=the output of theactuator) manipulated based on the manipulated variable contains steadydisturbance, the estimated value after the total dead time of the outputfrom the first detecting means represented by the data generated by theestimating means tends to suffer a steady error with respect to a truevalue. In such a situation, when the reference value is appropriatelychanged depending on the manipulated variable generated by themanipulated variable generating means, the accuracy of the estimatedvalue of the output from the first detecting means can be increased.

According to the second aspect of the present invention, thepredetermined reference value is variably set depending on themanipulated variable generated by the manipulated variable determiningmeans. In this manner, the accuracy of the estimated value of the outputfrom the first detecting means after the total dead time, which is thesum of the first dead time of the plant and the second dead time of theinput manipulating system, can be increased. It is thus possible toincrease the accuracy of the control process for converging the outputfrom the first detecting means to the target value while compensatingfor the effect of both the first dead time of the plant and the seconddead time of the input manipulating system.

According to the second aspect of the present invention, as with thefirst aspect of the present invention, the manipulated variablegenerated by the manipulated variable generating means may comprise atarget input to the plant, a target value for the plant differentialinput, or a corrective variable for the amount of operation of theactuator. Using the data representing the estimated value of the outputfrom the first detecting means, the manipulated variable may begenerated by a feedback control process for converging the estimatedvalue of the output from the first detecting means, which is representedby the data, to the target value. The actuator of the input manipulatingsystem generally contains a response delay as well as a dead time. Sucha response delay can be compensated for by the actuator control means.Therefore, the estimating means and the manipulated variable generatingmeans pose no problem even if any response delay in the model of theinput manipulating system is not taken into account.

In the second aspect of the present invention, the plant comprises anexhaust system ranging from a position upstream of a catalytic converterdisposed in an exhaust passage of an internal combustion engine forpurifying an exhaust gas from the internal combustion engine, to aposition downstream of the catalytic converter, the exhaust systemincluding the catalytic converter, the input to the plant comprising anair-fuel ratio of an air-fuel mixture combusted in the internalcombustion engine as the actuator for generating the exhaust gas whichenters the catalytic converter, the output from the plant comprising theconcentration of a component of the exhaust gas having passed throughthe catalytic converter.

Because the accuracy of the estimated value of the output (a detectedvalue of the concentration of the component) from the first detectingmeans represented by the data generated by the estimating means isincreased, the accuracy of the control process for converging the outputfrom the first detecting means to the target value can be increasedwhile compensating for the effect of both the dead time of the exhaustsystem, i.e., a time required until the air-fuel ratio at each point oftime given to the exhaust system is reflected in the output from thefirst detecting means, and the dead time of an input generating systemwith respect to the exhaust system, i.e., a system ranging from theinternal combustion engine to the actuator control means, i.e., a timerequired until the manipulated variable at each point of time given tothe input generating system is reflected in the actual air-fuel ratiogiven to the exhaust system. As a result, the purifying capability ofthe catalytic converter in the exhaust system is reliably achieved.

In this case, the manipulated variable generated by the manipulatedvariable generating means may comprise a target air-fuel ratio for theair-fuel mixture combusted in the internal combustion engine, a targetvalue for the difference (the plant differential input) between theair-fuel ratio of the air-fuel mixture and the reference value, or acorrective variable for the amount of fuel supplied to the internalcombustion engine. The actuator control means adjusts the amount of fuelsupplied to the internal combustion engine depending on the manipulatedvariable to manipulate the air-fuel ratio as the input given to theexhaust system (plant).

In the second aspect of the present invention, more specifically, thealgorithm comprises an algorithm for generating the data representingthe estimated value of the output from the first detecting means afterthe total dead time, using data representing the difference (plantdifferential output) between the output of the first detecting means andthe target value, data representing the manipulated variable, andparameters of the model of the plant which define the behavior of themodel of the plant.

If the model of the plant and the model of the input manipulating systemare combined with each other, then the combined system can be expressedby a model including the parameters of the model of the plant, as asystem for generating the plant differential output with the total deadtime and the response delay of the plant. Therefore, the datarepresenting the estimated value of the output from the first detectingmeans after the total dead time, specifically, the estimated value ofthe output from the first detecting means or the estimated value of theplant differential output, can be generated using the data of the plantdifferential output, the data of the manipulated variable, and theparameters of the model of the plant (parameters to be set to certainvalues in defining the behavior of the model of the plant).

The data representing the manipulated variable and used by theestimating means to generate the estimated value of the output from thefirst detecting means basically includes one or more past values of themanipulated variable prior to the second dead time of the inputmanipulating system. If the input to the plant (=the output of theactuator) is detected, then the manipulated variable at each point oftime prior to the second dead time can be represented by a detectedvalue prior to the present time of the input to the plant based on themodel of the input manipulating system.

In the second aspect of the present invention, the plant control systemfurther comprises second detecting means for detecting the input to theplant, and the data representing the manipulated variable and used bythe estimating means to generate the estimated value of the output fromthe first detecting means includes at least one past value of themanipulated variable prior to the second dead time of the inputmanipulating system, the algorithm comprising an algorithm forgenerating the estimated value of the output from the first detectingmeans using data representing the past value with an output from thesecond detecting means based on the model of the input manipulatingsystem, instead of all or some of past values of the manipulatedvariable prior to the second dead time.

By generating data representing the estimated value of the output fromthe first detecting means using data represented by the output from thesecond detecting means (corresponding to a detected value of the actualinput to the plant generated by the actuator depending on themanipulated variable), instead of all or some (preferably all) of pastvalues prior to the second dead time, of the data of the manipulatedvariable required to generate the data representing the estimated valueof the output from the first detecting means after the total dead time,the estimated value can be generated depending on the actual operatingstate of the actuator. As a result, the reliability of the data of theestimated value can be increased, and the accuracy of the controlprocess for converging the output from the first detecting means to thetarget value can further be increased.

Depending on the length of the second dead time, all the data of themanipulated variable required to generate the data representing theestimated value of the output from the first detecting means may becomeprior to the second dead time. In such a case, if all the data of themanipulated variable is replaced with data represented by the outputfrom the second detecting means, then the algorithm executed by theestimating means generates the data representing the estimated value ofthe output from the first detecting means after the total dead time,using the data of the plant differential output, the data represented bythe output from the second detecting means, and the parameters of themodel, rather than the data of the manipulated variable. The presentinvention covers such an aspect.

The plant control system according to the present invention which usesthe parameters of the model in order to generate the data representingthe estimated value of the output from the first detecting means hasidentifying means for sequentially identifying the parameters of themodel of the plant, using the data representing the difference (theplant differential output) between the output of the first detectingmeans and the target value, and the data representing the difference (adetected value of the plant differential output) between the output fromthe second detecting means and the reference value.

While the parameters of the model of the plant may be of constant values(fixed values), it is preferable to identify the parameters of the modelsequentially on a real-time basis for causing the model to match theactual behavior of the plant. The parameters of the model of the plantcan be identified by using the data of the plant differential output andthe data of the detected value of the plant differential input.

According to the finding of the inventors of the present application, ina situation where the output from the second detecting means suffers asteady offset or the input to the plant (=the output of the actuator)manipulated based on the manipulated variable contains steadydisturbance, if the reference value is constant as is conventional, thenthe identified values of the parameters of the model of the plant tendto suffer a steady error with respect to a true value, and hence theoutput from the first detecting means is also liable to suffer a steadyerror with respect to its target value. However, with the referencevalue being variably set depending on the manipulated variable accordingto the present invention, it is possible to increase the accuracy of theidentified values of the parameters of the model of the plant, and henceit is possible to increase the accuracy of the control process forconverging the output from the first detecting means to the target valuein various behavioral states of the plant. If the plant is the exhaustsystem of an internal combustion engine, then the purifying capabilityof a catalytic converter in the exhaust system is reliably achieved.

In the plant control systems according to the first and second aspects,the model of the plant can express the behavior of the plant with acontinuous system, specifically, a continuous-time system. However, themodel of the plant should preferably be such a model that the behaviorof the plant is expressed by a discrete system, specifically, adiscrete-time system.

With the model of the plant being constructed as a discrete system, itis easy to construct an algorithm for generating the data representingthe estimated value of the output from the first detecting means with acomputer in given control cycles. In particular, if the aboveidentifying means is provided, then the identifying means makes it easyto construct an algorithm for sequentially identifying the parameters ofthe model.

The model of the plant is a model (autoregressive model) where the plantdifferential output in each control cycle is expressed by the plantdifferential input in a past control cycle prior to the control cycleand the plant differential output.

When the model of the plant is expressed by an autoregressive model of adiscrete system, the parameters of the plant are coefficients relativeto the plant differential input and plant differential output in themodel.

With respect to the model of the input manipulating system according tothe second aspect of the present invention, the input manipulatingsystem may be expressed may occasionally be lowered. However, since thereference value is variably set depending on the manipulated variableaccording to the present invention, the quick response of theconvergence of the output from the first detecting means to the targetvalue can be increased. Thus, if the plant is the exhaust system of aninternal combustion engine, then the purifying capability of a catalyticconverter in the exhaust system is smoothly achieved.

When the manipulated variable is generated according to the feedbackcontrol process based on the model of the plant, if the model of theplant is expressed by a discrete-time system, then it is easy toconstruct an algorithm for generating the manipulated variable with acomputer in given control cycles.

In the plant control system which generates the manipulated variableaccording to the feedback control process based on the model of theplant, more specifically, the manipulated variable comprises a targetvalue for the difference (the plant differential input) between theinput to the plant and the reference value, the actuator control meanscomprising means for controlling operation of the actuator in order tomanipulate the input to the plant into a target input determined basedon the target value for the difference and the reference value.

That is, the manipulated variable comprises a target value for the plantdifferential input, and operation of the actuator is controlled in orderto manipulate as a system having only the second dead time. Therefore,it makes no substantial difference whether the model is expressed by acontinuous system or a discrete system.

In the plant control systems according to the first and second aspectsof the present invention, the manipulated variable generating meanscomprises means for generating the manipulated variable in order toconverge the output of the first detecting means to the target valueaccording to a feedback control process constructed based on the modelof the plant.

The estimating means can generate the manipulated variable using theestimated value of the output of the first detecting means according tovarious processes. If the manipulated variable is generated according toa feedback control process that requires the model of the object to becontrolled, e.g., a sliding mode control process, then the feedbackcontrol process can be constructed on the basis of the plant.

According to various studies made by the inventors of the presentapplication, when the reference value relative to the plant differentialinput as the input to the plant is constant as is conventional, if theinput to the plant (=the output of the actuator) manipulated based onthe manipulated variable contains steady disturbance or the output fromthe second detecting means suffers a steady offset, then the quickresponse of the convergence of the output from the first detecting meansto the target value the actual input to the plant into a target inputdetermined based on the target value for the plant differential inputand the reference value. Thus, it is made easy to construct the feedbackcontrol process based on the model of the plant, and operation of theactuator based on the manipulated variable (the target value for theplant differential input) can adequately be controlled.

In manipulating the input to the plant into the target input with theactuator control means, if the plant control system has the seconddetecting means for detecting the input to the plant for the processingoperation of the estimating means, then operation of the actuator shouldpreferably be controlled according to the feedback control process inorder to converge the output from the second detecting means (a detectedvalue of the input to the plant) to the target input. Even if thedetected value of the input to the plant is not used for the processingoperation of the estimating means, it is preferable to provide thesecond detecting means and to control operation of the actuatoraccording to the feedback control process. In this fashion, the input tothe plant can appropriately be controlled at the target air-fuel ratio.

However, when an amount of operation of the actuator, specifically, theamount of fuel supplied to an internal combustion engine as theactuator, or its corrective variable is determined from the target inputusing a map or the like, the output of the actuator which is the inputto the plant can be manipulated into the target input according to afeed-forward control process.

When operation of the actuator is controlled according to the feedbackcontrol process, the feedback control process is preferably carried outby a recursive-type controller such as an adaptive controller.Specifically, the recursive-type controller determines a new feedbackmanipulated variable according to a recursive formula containing a givennumber of time-series data prior to the present of a feedbackmanipulated variable for operation of the actuator (e.g., a correctivevariable for the amount of operation of the actuator, a correctivevariable for the amount of fuel supplied to an internal combustionengine as the actuator, etc.). With the recursive-type controller beingused, the feedback control process for converging the output from thesecond detecting means (a detected value of the input to the plant) tothe target input while compensating for the effect of a response delayof the actuator.

In the plant control system which generates the manipulated variablewith the manipulated variable generating means according to the feedbackcontrol process based on the model of the plant, the feedback controlprocess comprises a process of generating the manipulated variable usingdata representing the difference between the estimated value of theoutput from the first detecting means represented by the data generatedby the estimating means and the target value, and parameters of themodel of the plant which define the behavior of the model of the plant.

By using the data representing the difference between the estimatedvalue of the output from the first detecting means and the target value(which difference corresponds to the estimated value of the plantdifferential output after the dead time of the plant according to thefirst aspect, and the estimated value of the plant differential outputafter the total dead time of the plant according to the second aspect)and the parameters of the model of the plant which define the behaviorof the model, it is possible to generate an appropriate manipulatedvariable required to converge the output from the first detecting meansto the target value while compensating for the dead time of the plant,or the total dead time of both the plant and the input manipulatingsystem.

If the parameters of the model of the plant are identified by theidentifying means, in particular, then the manipulated variable is madeadequate in various behavioral states of the plant. By variably settingthe reference value, as described above, the accuracy of the estimatedvalue of the output from the first detecting means, and the accuracy ofthe identified values of the parameters of the model of the plant can beincreased, and the quick response of the convergence of the output fromthe first detecting means to the target value can be increased.Therefore, the control process for converging the output from the firstdetecting means to the target value can be performed highly accuratelywith a highly quick response. Thus, if the plant is the exhaust systemof an internal combustion engine, then the purifying capability of acatalytic converter in the exhaust system is reliably and smoothlyachieved.

In the feedback control process (e.g., the sliding mode control process)performed for the manipulated variable generating means to generate themanipulated variable, it is possible to construct a simple algorithmwhich does not use the parameters of the plant.

In the plant control system for generating the manipulated variableaccording to the feedback control process based on the model of theplant, the feedback control process should preferably be the slidingmode control process, and the sliding mode control process shouldpreferably be the adaptive sliding mode control process.

The sliding mode control process is a variablestructure feedback controlprocess, and is stabler against disturbances than the known PID controlprocess. The adaptive sliding mode control process is a combination ofan ordinary sliding mode control process and a control law referred toas an adaptive control law for eliminating the effect of disturbances asmuch as possible. By generating the manipulated variable according tothe sliding mode control process, preferably, the adaptive sliding modecontrol process, the stability of the control process for converging theoutput from the first detecting means to the target value is increased.

If the feedback control process comprises the sliding mode controlprocess, then the sliding mode control process generates the targetvalue for the plant differential input as the manipulated variable.

If the feedback control process comprises the adaptive sliding modecontrol process, then the manipulated variable generated by themanipulated variable generating means according to the adaptive slidingmode control process includes an adaptive control law component based onan adaptive control law (adaptive algorithm) of the adaptive slidingmode control process. The reference value variable setting meanscomprises means for variably setting the reference value based on thevalue of the adaptive control law component of the manipulated variable.

The manipulated variable generated according to the adaptive slidingmode control process is given as the sum of a component based on acontrol law (so-called equivalent control input) for holding(restricting) the value of a function referred to as a switchingfunction used in the adaptive sliding mode control process to “0”, acomponent based on a reaching control law for converging the value ofthe switching function to “0”, and a component based on an adaptivecontrol law for eliminating disturbances as much as possible inconverging the value of the switching function to “0”.

The switching function is represented by a linear function or the likehaving as a component time-series data of the difference between acontrolled variable (which is the output from the first detecting meansin this invention) and a target value. The component based on theadaptive control law is of a value proportional to an integrated value(time integral) of the value of the switching function, for example.

According the finding of the inventors of the present application, byvariably setting the reference value depending on the component based onthe adaptive control law (adaptive control law component) in the plantcontrol system, the effect of disturbance steadily contained in theoutput of the actuator which is the input to the plant, more generally,the effect of an error of various data required to generate themanipulated variable with respect to a true value, can appropriately becompensated for. As a result, the accuracy and quick response of thecontrol process for converging the output from the first detecting meansto the target value can appropriately be increased.

Specifically, the reference value variable setting means comprises meansfor variably setting the reference value by increasing or decreasing thereference value depending on the magnitude of the value of the adaptivecontrol law component of the manipulated variable with respect to apredetermined value or a range close to and containing the predeterminedvalue.

The reference value is variably set such that the value of the adaptivecontrol law component is converged to the predetermined value or a valueclose thereto. In this manner, the reference value can be variably setappropriately depending on the value of the adaptive control lawcomponent. The reference value serves as a reference for the plantdifferential input as the input to the model of the plant on which theadaptive sliding mode control process is based. If the reference valueis varied too frequently, it will adversely affect the manipulatedvariable generated according to the adaptive sliding mode controlprocess, tending to impair the stability of the output from the firstdetecting means. Therefore, in variably setting the reference valuedepending on the value of the adaptive control law component, it ispreferable to increase or decrease the reference value depending on themagnitude of the value of the adaptive control law component withrespect to the predetermined value or the range close to and containingthe predetermined value (the reference value is not varied when thevalue of the adaptive control law component is present in the aboverange).

When the feedback control process for generating the manipulatedvariable comprises the sliding mode control process (including theadaptive sliding mode control process), the reference value variablesetting means preferably comprises means for sequentially determiningwhether the output from the first detecting means is stable or not, andholding the reference value as a predetermined value irrespective of themanipulated variable if the output from the first detecting means isunstable.

Specifically, the reference value affects the manipulated variablegenerated according to the sliding mode control process, and hence theoutput from the first detecting means. Consequently, in a situationwhere the output from the first detecting means is judged as unstable,the reference value is not variably set, but is held to a predeterminedvalue (e.g., the present value or a predetermined fixed value). In thismanner, an inappropriate manipulated variable which will make the outputfrom the first detecting means more unstable can be avoided.

The process of holding the reference value to the predetermined value inthe situation where the output from the first detecting means is judgedas unstable is preferably carried out when the reference value isvariably set depending on the value of the adaptive control lawcomponent.

The reference value variable setting means comprises means fordetermining whether the output from the first detecting means is stableor not based on the value of a switching function used in the slidingmode control process.

In the sliding mode control process, it is important to converge thevalue of the switching function stably to “0” in stably converging thecontrolled variable (which is the output from the first detecting meansin this invention) to the target value. Thus, the stability of theoutput from the first detecting means can be determined based on thevalue of the switching function.

For example, when the product of the value of the switching function andits rate of change (which product corresponds to the time-differentiatedvalue of a Lyapunov function) is determined, if the product is of apositive value, then the value of the switching function is getting awayfrom “0”, and if the product is of a negative value, then the value ofthe switching function is getting closely to “0”. Basically, therefore,it is possible to determine whether the output from the first detectingmeans is unstable or stable depending on whether the above product is ofa positive value or a negative value. Instead, the stability of theoutput from the first detecting means can be determined by comparing themagnitude of the value of the switching function or the magnitude of itsrate of change with a suitable given value.

In the plant control system, the reference value variable setting meanscomprises means for determining whether the output from the firstdetecting means is substantially converged to the target value or not,and holding the reference value as a predetermined value irrespective ofthe manipulated variable if the output from the first detecting means isnot converged to the target value.

Specifically, in a situation where the output from the first detectingmeans is not converged to the target value, the generated state of themanipulated variable tends to be unstable. Therefore, the referencevalue is not variably set, but is held to a predetermined value (e.g.,the present value or a predetermined fixed value).

The generated state of the manipulated variable is therefore made stableas much as possible, allowing the output from the first detecting meansto be converged stably to the target value. If the plant is the exhaustsystem of an internal combustion engine, then the purifying capabilityof a catalytic converter in the exhaust system is stably achieved.

The process of holding the reference value to the predetermined value inthe situation where the output from the first detecting means is notconverged to the target value is preferably carried out when thereference value is variably set depending on the value of the adaptivecontrol law component relative to the sliding mode control process.Whether the output from the first detecting means is substantiallyconverged to the target value or not can be determined by comparing themagnitude of the difference (plant differential output) of the outputfrom the first detecting means and the target value with a suitablegiven value. More specifically, if the magnitude of the difference isequal to or smaller than the given value, then the output from the firstdetecting means is judged as being substantially converged to the targetvalue, and if the magnitude of the difference is greater than the givenvalue, then the output from the first detecting means is judged as beingnot converged to the target value.

If the plant is the exhaust system of an internal combustion engine,then in achieving optimum the purifying capability of a catalyticconverter in the exhaust system, it is preferable to use an O₂ sensor(oxygen concentration sensor) as the first detecting means and to set atarget value for the output of the O₂ sensor to a predetermined constantvalue.

The above and other objects, features, and advantages of the presentinvention will become apparent from the following description when takenin conjunction with the accompanying drawings which illustrate preferredembodiments of the present invention by way of example.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a plant control system according to anembodiment of the present invention;

FIG. 2 is a diagram showing output characteristics of an O₂ sensor usedin the plant control system shown in FIG. 1;

FIG. 3 is a block diagram showing a basic arrangement of a targetair-fuel ratio generator in the plant control system shown in FIG. 1;

FIG. 4 is a diagram illustrative of a sliding mode control processemployed by the plant control system shown in FIG. 1;

FIG. 5 is a block diagram of an adaptive controller in the plant controlsystem shown in FIG. 1;

FIG. 6 is a flowchart of an engine operation control process carried outby the plant control system shown in FIG. 1;

FIG. 7 is a flowchart of a subroutine of the flowchart shown in FIG. 6;

FIG. 8 is a flowchart of an overall process carried out by the targetair-fuel ratio generator in the plant control system shown in FIG. 1;

FIG. 9 is a flowchart of a subroutine of the flowchart shown in FIG. 8;

FIG. 10 is a flowchart of a subroutine of the flowchart shown in FIG. 8;

FIG. 11 is a diagram illustrative of partial processing of the flowchartshown in FIG. 10;

FIG. 12 is a diagram illustrative of partial processing of the flowchartshown in FIG. 10;

FIG. 13 is a flowchart of a subroutine of the flowchart shown in FIG.10;

FIG. 14 is a flowchart of a subroutine of the flowchart shown in FIG. 8;

FIG. 15 is a flowchart of a subroutine of the flowchart shown in FIG. 8;

FIG. 16 is a flowchart of a subroutine of the flowchart shown in FIG. 8;

FIG. 17 is a diagram illustrative of the flowchart shown in FIG. 16;

FIG. 18 is a diagram illustrative of an allowable range for a limitingprocess carried out in a subroutine of the flowchart shown in FIG. 8;

FIG. 19 is a flowchart of a subroutine of the flowchart shown in FIG. 8;

FIG. 20 is a flowchart of a subroutine of the flowchart shown in FIG.19;

FIG. 21 is a diagram illustrative of the flowchart shown FIG. 19;

FIG. 22 is a diagram illustrative of the flowchart shown FIG. 19;

FIG. 23 is a diagram illustrative of the flowchart shown FIG. 19;

FIG. 24 is a diagram illustrative of the flowchart shown FIG. 19;

FIG. 25 is a diagram illustrative of the flowchart shown FIG. 19;

FIG. 26 is a flowchart of a subroutine of the flowchart shown in FIG. 8;

FIG. 27 is a diagram illustrative of the flowchart shown FIG. 26; and

FIG. 28 is a block diagram of a plant control system according to asecond embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

A plant control system according to an embodiment of the presentinvention will be described below with reference to FIGS. 1 through 27.According to this embodiment, the plant control system is a system forcontrolling a plant which comprises an exhaust system of an internalcombustion engine that ranges from a position upstream of a catalyticconverter for purifying an exhaust gas emitted from the internalcombustion engine to a position downstream of the catalytic converter.

FIG. 1 shows in block form the plant control system according to theembodiment. As shown in FIG. 1, a four-cylinder internal combustionengine 1 mounted on a propulsion source on an automobile or a hybridvehicle, i.e., a drive source for drive wheels thereof. The internalcombustion engine 1 generates exhaust gases produced by combustion of anair-fuel mixture in the cylinders. The exhaust gases are collected intoa common discharge pipe 2 (exhaust passage) positioned near the internalcombustion engine 1, from which the exhaust gas is discharged into theatmosphere. Two catalytic converters 3, 4, each comprising a three-waycatalytic converter combined with a NOx absorber (nitrogen oxideabsorber), are mounted in the common exhaust pipe 2 at successivelydownstream locations thereon.

The NOx absorber in each of the catalytic converters 3, 4 may be of theocclusion type which occludes NOx when the air-fuel ratio of theair-fuel mixture combusted in the internal combustion engine 1 is of alean state (the fuel is lesser than the stoichiometric air-fuel ratio)and the concentration of oxygen in the exhaust gas is relatively high(NOx in the exhaust gas is relatively high), or of the adsorption typewhich adsorbs NOx in the exhaust gas when the air-fuel ratio of theair-fuel mixture is of a lean state. The occlusion-type NOx absorber maybe made of barium oxide (BaO), and the adsorption-type NOx absorber maybe made of sodium (Na), titanium (Ti), or strontium (Sr).

When the air-fuel ratio of the air-fuel mixture combusted in theinternal combustion engine 1 is close to the stoichiometric air-fuelratio or of a rich state (the fuel is greater than the stoichiometricair-fuel ratio), making the oxygen concentration in the exhaust gasrelatively low, the occlusion-type NOx absorber discharges the occludedNOx, which is reduced by HC (hydrocarbon) and CO (carbon monoxide) inthe exhaust gas. When the air-fuel ratio of the air-fuel mixturecombusted in the internal combustion engine 1 is close to thestoichiometric air-fuel ratio or of a rich state (the fuel is greaterthan the stoichiometric air-fuel ratio), making the oxygen concentrationin the exhaust gas relatively low, the adsorption-type NOx absorbercauses the adsorbed NOx to be reduced by HC (hydrocarbon) and CO (carbonmonoxide) in the exhaust gas, producing a nitrogen gas that isdischarged from the NOx absorber.

Of the catalytic converters 3, 4, the catalytic converter related to thepresent invention is the upstream catalytic converter 3, and thedownstream catalytic converter 4 may be dispensed with. The internalcombustion engine 1 corresponds to an actuator.

The plant control system serves to control an air-fuel ratio of theinternal combustion engine 1 (more accurately, the air-fuel ratio of anair-fuel mixture combusted by the internal combustion engine 1) in orderto enable the catalytic converter 3 to achieve optimum exhaust gaspurifying performance. The plant control system comprises a wide-rangeair-fuel ratio sensor 5 mounted on the exhaust pipe 2 upstream of thecatalytic converter 3, or more precisely at a position where exhaustgases from the cylinders of the internal combustion engine 1 are puttogether, an O₂ sensor (oxygen concentration sensor) 6 mounted on theexhaust pipe 2 downstream of the catalytic converter 3 and upstream ofthe catalytic converter 4, and a control unit 7 for carrying out acontrol process (described later on) based on detected output signalsfrom the sensors 5, 6. The control unit 7 is supplied with detectedoutput signals from the sensors 5, 6 and also detected output signalsfrom various other sensors for detecting operating conditions of theinternal combustion engine 1, including a engine speed sensor, an intakepressure sensor, a coolant temperature sensor, etc. The wide-rangeair-fuel ratio sensor 5 and the O₂ sensor 6 correspond respectively to asecond detecting means and a first detecting means.

The wide-range air-fuel ratio sensor 5 is in the form of an O₂ sensor,and outputs a signal having a level depending on the air-fuel ratio ofan air-fuel mixture from which an exhaust gas introduced into thecatalytic converter 3 is generated by fuel combustion in the internalcombustion engine 1. The air-fuel ratio is represented by the oxygenconcentration of the exhaust gas introduced into the catalyticconverter. The output signal from the wide-range air-fuel ratio sensor 5(hereinafter referred to as an LAF sensor 5) is processed by a detectingcircuit such as a linearizer (not shown) into a signal having an outputsignal KACT having a level which is proportional to the air-fuel ratioof the air-fuel mixture combusted in the internal combustion engine 1 ina wide range of air-fuel ratios, i.e., an output signal KACTrepresentative of a detected value of the air-fuel ratio. The LAF sensor5 is disclosed in detail in Japanese laid-open patent publication No.4-369471, which corresponds to U.S. Pat. No. 5,391,282, and will not bedescribed below.

The O₂ sensor 6 disposed downstream of the catalytic converter 3generates an output signal VO2/OUT having a level depending on theoxygen concentration in the exhaust gas that has passed through thecatalytic converter 3, i.e., an output signal VO2/OUT representing adetected value of the oxygen concentration in the exhaust gas, as withordinary O₂ sensors. The output signal VO2/OUT from the O₂ sensor 6 willchange with high sensitivity in substantial proportion to the oxygenconcentration in the exhaust gas that has passed through the catalyticconverter 3, with the air-fuel ratio recognized by the oxygenconcentration in the exhaust gas being in a range Δ close to astoichiometric air-fuel ratio, as shown in FIG. 2.

The control unit 7 comprises a microcomputer and is roughly divided intoa control unit 7 a for performing in predetermined control cycles aprocess for successively generating a target air-fuel ratio KCMD (whichis also a target value for the output KACT of the LAF sensor 5) for theinternal combustion engine 1, and a control unit 7 b for performing inpredetermined control cycles a process for controlling the air-fuelratio of the internal combustion engine 1 at the target air-fuel ratioKCMD. The control unit 7 a will be referred to as an exhaust-sidecontrol unit 7 a, and the control unit 7 b will be referred to as anengineside control unit 7 b.

The engine-side control unit 7 b corresponds to an actuator controlmeans.

The target air-fuel ratio KCMD generated by the exhaust-side controlunit 7 a is basically an air-fuel ratio of the internal combustionengine 1 required to set the output of the O₂ sensor 6 (the detectedvalue of the oxygen concentration) at a predetermined target value(constant value). The control cycles of the process performed by theexhaust-side control unit 7 a for generating the target air-fuel ratioKCMD are of a constant period (e.g., 30-100 ms) in view of a relativelylong dead time of an exhaust system E (described later on) including thecatalytic converter 3 and calculating loads.

The process performed by the engine-side control unit 7 b forcontrolling the air-fuel ratio of the internal combustion engine 1, morespecifically the process for adjusting the amount of fuel supplied tothe internal combustion engine 1, is required to be synchronous with therotational speed of the internal combustion engine 1. The control cyclesof the process performed by the engine-side control unit 7 b are of aperiod in synchronism with a crankshaft angle period (so-called TDC) ofthe internal combustion engine 1. Output data from various sensorsincluding the LAF sensor 5 and the O₂ sensor 6 are also read insynchronism with the crankshaft angle period (TDC) of the internalcombustion engine 1.

The constant period of the control cycles of the exhaust-side controlunit 7 a is longer than the crankshaft angle period (TDC).

The engine-side control unit 7 b has, as its main functional components,a basic fuel injection quantity calculator 8 for determining a basicfuel injection quantity Tim to be injected into the internal combustionengine 1, a first correction coefficient calculator 9 for determining afirst correction coefficient KTOTAL to correct the basic fuel injectionquantity Tim, and a second correction coefficient calculator 10 fordetermining a second correction coefficient KCMDM to correct the basicfuel injection quantity Tim.

The basic fuel injection quantity calculator 8 determines a referencefuel injection quantity for the internal combustion engine 1 from therotational speed NE and intake pressure PB of the internal combustionengine 1 using a predetermined map, and corrects the determinedreference fuel injection quantity depending on the effective openingarea of a throttle valve (not shown) of the internal combustion engine1, thereby calculating a basic fuel injection quantity Tim.

The first correction coefficient KTOTAL determined by the firstcorrection coefficient calculator 9 serves to correct the basic fuelinjection quantity Tim in view of an exhaust gas recirculation ratio ofthe internal combustion engine 1, i.e., the proportion of an exhaust gascontained in an air-fuel mixture introduced into the internal combustionengine 1, an amount of purged fuel supplied to the internal combustionengine 1 when a canister (not shown) is purged, a coolant temperature,an intake temperature, etc. of the internal combustion engine 1.

The second correction coefficient KCMDM determined by the secondcorrection coefficient calculator 10 serves to correct the basic fuelinjection quantity Tim in view of the charging efficiency of an air-fuelmixture due to the cooling effect of fuel flowing into the internalcombustion engine 1 depending on a target air-fuel ratio KCMD determinedby the exhaust-side control unit 7 a (described later on).

The control unit 7 corrects the basic fuel injection quantity Tim withthe first correction coefficient KTOTAL and the second correctioncoefficient KCMDM by multiplying the basic fuel injection quantity Timby the first correction coefficient KTOTAL and the second correctioncoefficient KCMDM, thus producing a demand fuel injection quantity Tcylfor the internal combustion engine 1.

Specific details of processes for calculating the basic fuel injectionquantity Tim, the first correction coefficient KTOTAL, and the secondcorrection coefficient KCMDM are disclosed in detail in Japaneselaid-open patent publication No. 5-79374 and U.S. Pat. No. 5,253,630,and will not be described below.

The engine-side control unit 7 b also has, in addition to the abovefunctional components, a feedback controller 14 for adjusting a fuelinjection quantity of the internal combustion engine 1 according to afeedback control process so as to converge the output signal KACT of theLAF sensor 5 (the detected air-fuel ratio) to the target air-fuel ratioKCMD which is sequentially determined by the exhaust control unit 7 a.

The feedback controller 14 comprises a general feedback controller 15for feedback-controlling a total air-fuel ratio for all the cylinders ofthe internal combustion engine 1 and a local feedback controller 16 forfeedback-controlling an air-fuel ratio for each of the cylinders of theinternal combustion engine 1.

The general feedback controller 15 sequentially determines a feedbackcorrection coefficient KFB to correct the demand fuel injection quantityTcyl (by multiplying the demand fuel injection quantity Tcyl) so as toconverge the output signal KACT from the LAF sensor 5 to the targetair-fuel ratio KCMD.

The general feedback controller 15 comprises a PID controller 17 fordetermining a feedback manipulated variable KLAF as the feedbackcorrection coefficient KFB depending on the difference between theoutput signal KACT from the LAF sensor 5 and the target air-fuel ratioKCMD according to a known PID control process, and an adaptivecontroller 18 (indicated by “STR” in FIG. 1) for adaptively determininga feedback manipulated variable KSTR for determining the feedbackcorrection coefficient KFB in view of changes in operating conditions ofthe internal combustion engine 1 or characteristic changes thereof fromthe output signal KACT from the LAF sensor 5 and the target air-fuelratio KCMD.

In present embodiment, the feedback manipulated variable KLAF generatedby the PID controller 17 is of “1” and can be used directly as thefeedback correction coefficient KFB when the output signal KACT (thedetected air-fuel ratio) from the LAF sensor 5 is equal to the targetair-fuel ratio KCMD. The feedback manipulated variable KSTR generated bythe adaptive controller 18 becomes the target air-fuel ratio KCMD whenthe output signal KACT from the LAF sensor 5 is equal to the targetair-fuel ratio KCMD. A feedback manipulated variable kstr (=KSTR/KCMD)which is produced by dividing the feedback manipulated variable KSTR bythe target air-fuel ratio KCMD with a divider 19 can be used as thefeedback correction coefficient KFB.

The feedback manipulated variable KLAF generated by the PID controller17 and the feedback manipulated variable kstr which is produced bydividing the feedback manipulated variable KSTR from the adaptivecontroller 18 by the target air-fuel ratio KCMD are selected one at atime by a switcher 20. A selected one of the feedback manipulatedvariable KLAF and the feedback manipulated variable KSTR is used as thefeedback correction coefficient KFB. The demand fuel injection quantityTcyl is corrected by being multiplied by the feedback correctioncoefficient KFB. Details of the general feedback controller 15(particularly, the adaptive controller 18) will be described later on.

The local feedback controller 16 comprises an observer 21 for estimatingreal air-fuel ratios #nA/F (n=1, 2, 3, 4) of the respective cylindersfrom the output signal KACT from the LAF sensor 5, and a plurality ofPID controllers 22 (as many as the number of the cylinders) fordetermining respective feedback correction coefficients #nKLAF for fuelinjection quantities for the cylinders from the respective real air-fuelratios #nA/F estimated by the observer 21 according to a PID controlprocess so as to eliminate variations of the air-fuel ratios of thecylinders.

Briefly stated, the observer 21 estimates a real air-fuel ratio #nA/F ofeach of the cylinders as follows: A system from the internal combustionengine 1 to the LAF sensor 5 (where the exhaust gases from the cylindersare combined) is considered to be a system for generating an air-fuelratio detected by the LAF sensor 5 from a real air-fuel ratio #nA/F ofeach of the cylinders, and is modeled in view of a detection responsedelay (e.g., a time lag of first order) of the LAF sensor 5 and achronological contribution of the air-fuel ratio of each of thecylinders to the air-fuel ratio detected by the LAF sensor 5. Based onthe modeled system, a real air-fuel ratio #nA/F of each of the cylindersis estimated from the output signal KACT from the LAF sensor 5.

Details of the observer 21 are disclosed in Japanese laid-open patentpublication No. 7-83094 and U.S. Pat. No. 5,531,208, for example, andwill not be described below.

Each of the PID controllers 22 of the local feedback controller 16divides the output signal KACT from the LAF sensor 5 by an average valueof the feedback correction coefficients #nKLAF determined for all thecylinders by the respective PID controllers 22 in a preceding controlcycle to produce a quotient value, and uses the quotient value as atarget air-fuel ratio for the corresponding cylinder. Each of the PIDcontrollers 22 then determines a feedback correction coefficient #nKLAFfor each cylinder in a present control cycle so as to eliminate anydifference between the target air-fuel ratio and the corresponding realair-fuel ratio #nA/F determined by the observer 21.

The local feedback controller 16 multiplies a value, which has beenproduced by multiplying the demand fuel injection quantity Tcyl by theselected feedback correction coefficient KFB produced by the generalfeedback controller 15, by the feedback correction coefficient #nKLAFfor each of the cylinders, thereby determining an output fuel injectionquantity #nTout (n=1, 2, 3, 4) for each of the cylinders.

The output fuel injection quantity #nTout thus determined for each ofthe cylinders is corrected for accumulated fuel particles on intake pipewalls of the internal combustion engine 1 by a corresponding fuelaccumulation corrector 23 in the engine-side control unit 7 b. Thecorrected output fuel injection quantity #nTout is applied to each offuel injectors (not shown) of the internal combustion engine 1, whichinjects fuel into each of the cylinders with the corrected output fuelinjection quantity #nTout.

The correction of the output fuel injection quantity in view ofaccumulated fuel particles on intake pipe walls is disclosed in detailin Japanese laid-open patent publication No. 8-21273 and U.S. Pat. No.5,568,799, for example, and will not be described in detail below.

A sensor output selector 24 shown in FIG. 1 serves to select the outputsignal KACT from the LAF sensor 5, which is suitable for the estimationof a real air-fuel ratio #nA/F of each cylinder with the observer 21,depending on the operating conditions of the internal combustion engine1. Details of the sensor output selector 24 are disclosed in detail inJapanese laid-open patent publication No. 7-259588 and U.S. Pat. No.5,540,209, and will not be described in detail below.

The exhaust-side control unit 7 a has, as its main functional component,a target air-fuel ratio generator 28 for sequentially generating thetarget air-fuel ratio KCMD in control cycles of the engine-side controlunit 7 b using the data of the output signal KACT of the LAF sensor 5which is given via the engine-side control unit 7 a and the data of theoutput signal VO2/OUT of the O₂ sensor 6.

As shown in FIG. 3, the target air-fuel ratio generator 28 comprises areference value setting unit 11 (reference value variable setting means)for sequentially variably setting a reference value FLAF/BASE withrespect to the air-fuel ratio of the internal combustion engine 1(hereinafter referred to as an “air-fuel ratio reference valueFLAF/BASE”, which is also a reference value with respect to the outputsignal KACT of the LAF sensor 5), a subtractor 12 for determining adifference kact (=KACT−FLAF/BASE) between the output signal KACT fromthe LAF sensor 5 and the air-fuel ratio reference value FLAF/BASE, and asubtractor 13 for determining a difference VO2 (=VO2/OUT−VO2/TARGET)between the output signal VO2/OUT from the O₂ sensor 6 and a targetvalue VO2/TARGET therefor. The target value VO2/TARGET for the outputsignal VO2/OUT from the O₂ sensor 6 is established as a constant value(fixed value) since in the present embodiment, optimum exhaust gaspurifying performance of the catalytic converter 3 can be achieved withthe output signal VO2/OUT from the O₂ sensor 6 being set at a certainconstant value (see FIG. 2).

The differences kact, VO2 determined respectively by the subtractors 12,13 are referred to as a differential output kact of the LAF sensor 5 anda differential output VO2 of the O₂ sensor 6, respectively.

The target air-fuel ratio generator 28 also has a manipulated variablegenerator 29 for sequentially generating a manipulated variable usl forcontrolling the airfuel ratio of the internal combustion engine 1 inorder to converge the output signal VO2/OUT from the O₂ sensor 6 to thetarget value VO2/TARGET, i.e., to converge the differential output VO2to “0”, using the data of the differential output kact of the LAF sensor5 and the differential output VO2 of the O₂ sensor 6, a limiter 30 forgenerating a manipulated variable kcmd by limiting the manipulatedvariable usl to a value within a predetermined allowable range, and anadder 31 for adding the air-fuel ratio reference value FLAF/BASEestablished by the reference value setting unit 11 to the manipulatedvariable kcmd generated by the limiter 30.

The manipulated variable generator 29 corresponds to a manipulatedvariable generating means.

The manipulated variable usl generated by the manipulated variablegenerator 29 as described later on represents the difference between theactual air-fuel ratio (detected by the LAF sensor 5) of the internalcombustion engine 1 required to converge the output signal VO2/OUT fromthe O₂ sensor 6 to the target value VO2/TARGET, and the air-fuel ratioreference value FLAF/BASE. Therefore, basically, in order to convergethe output signal VO2/OUT from the O₂ sensor 6 to the target valueVO2/TARGET, the sum of the manipulated variable usl (hereinafterreferred to as a “demand differential air-fuel ratio usl”) and theair-fuel ratio reference value FLAF/BASE may be generated as the targetair-fuel ratio KCMD.

The demand differential air-fuel ratio usl generated by the manipulatedvariable generator 29 may occasionally suffer relatively large changesdue to disturbances. When the actual air-fuel ratio (detected by the LAFsensor 5) of the internal combustion engine 1 is controlled at thetarget air-fuel ratio (=usl+FLAF/BASE) determined depending on thedemand differential air-fuel ratio usl, the internal combustion engine 1may possibly operate unstably.

In order to avoid the above drawback, the limiter 30 in the targetair-fuel ratio generator 28 limits the demand differential air-fuelratio usl to generate the manipulated variable kcmd whose value islimited within the allowable range.

The limiter 30 limits the demand differential air-fuel ratio usl asfollows: When the value of the demand differential air-fuel ratio usl isin the allowable range (a normal state), the limiter 30 sets up thedemand differential air-fuel ratio usl as the manipulated variable kcmd.When the value of the demand differential air-fuel ratio usl deviatesfrom the allowable range beyond its upper limit value or lower limitvalue, the limiter 30 forcibly sets the demand differential air-fuelratio usl to the upper limit value or lower limit value of the allowablerange.

The adder 31 in the target air-fuel ratio generator 28 adds the air-fuelratio reference value FLAF/BASE to the manipulated variable kcmd(hereinafter referred to as a “command differential air-fuel ratiokcmd”) produced by limiting the demand differential air-fuel ratio usl,generating the target air-fuel ratio KCMD, i.e., the target air-fuelratio KCMD (=kcmd+FLAF/BASE) to be given as a command value for theair-fuel ratio of the internal combustion engine 1 to the engine-sidecontrol unit 7 b.

The manipulated variable generator 29 will further be described below.The manipulated variable generator 29 sequentially determines the demanddifferential air-fuel ratio usl in control cycles of the exhaust-sidecontrol unit 7 a as a control input go be given to an exhaust system E(hereinafter referred to as an “object exhaust system E”), which extendsfrom the position of the LAF sensor 5 in the exhaust pipe 2 (upstream ofthe catalytic converter 3) to the position of the O₂ sensor 6(downstream of the catalytic converter 3) and which includes thecatalytic converter 3, in order to converge the output signal VO2/OUT ofthe O₂ sensor 6 to the target value VO2/TARGET therefor, i.e., toconverge the differential output VO2 of the O₂ sensor 6 to “0”,according to a sliding mode control process, which is a type of feedbackcontrol process, in view of a dead time present in the object exhaustsystem E, dead times present in the internal combustion engine 1 and theengine control unit 7 b, and behavioral changes of the object exhaustsystem E.

The object exhaust system E corresponds to a plant.

For generating the demand differential air-fuel ratio usl, the objectexhaust system E is regarded as a system for generating the differentialoutput VO2 of the O₂ sensor 6 with a dead time and a response delay fromthe differential output kact of the LAF sensor 5 which corresponds tothe difference between the actual air-fuel ratio of the air-fuelmixture, i.e., the air-fuel mixture from which the exhaust gas enteringthe catalytic converter 3 is generated upon combustion, and the air-fuelratio reference value FLAF/BASE, and the behavior of the system ismodeled in advance. In addition, the system comprising the internalcombustion engine 1 and the engine-side control unit 7 b is regarded asa system (hereinafter referred to as an “air-fuel ratio manipulatingsystem”) for generating the differential output kact of the LAF sensor 5with a dead time from the command differential air-fuel ratio kcmd(which usually agrees with the demand differential air-fuel ratio usl)that represents the difference (=KCMD−FLAF/BASE) between the targetair-fuel ratio KCMD and the air-fuel ratio reference value FLAF/BASE,i.e., a system in which the differential output kact of the LAF sensor 5at each point of time agrees with the command differential air-fuelratio kcmd prior to the dead time of the air-fuel ratio manipulatingsystem, and the behavior of the system is modeled in advance.

The model representing the behavior of the object exhaust system E(hereinafter referred to as an “exhaust system model”) is expressed as amodel of a discrete-time system (more specifically, an autoregressivemodel having a dead time in the differential output kact as an input tothe object exhaust system E) according to the following equation (1):

VO2(k+1)=a1·VO2(k)+a2·VO2(k−1)+b1·kact(k−d1)  (1)

where “k” represents the number of a discrete-time control cycle of theexhaust-side control unit 7 a, and “d1” the dead time present in theobject exhaust system E as expressed in terms of the number of controlcycles. The dead time of the object exhaust system E (the time requireduntil the air-fuel ratio detected by the LAF sensor 5 at each point oftime is reflected in the output signal VO2/OUT from the O₂ sensor 6) isgenerally equal to the time of 3−10 control cycles (d1=3−10) if theperiod (constant) of control cycles of the exhaust-side control unit 7 aranges from 30 to 100 ms. In present embodiment, a preset constant value(e.g., d1=7 in the present embodiment) equal to or slightly longer thanthe actual dead time of the object exhaust system E is used as the deadtime d1 in the exhaust system model as represented by the equation (1).

The first and second terms of the right side of the equation (1)correspond to a response delay of the object exhaust system E, the firstterm being a primary autoregressive term and the second term being asecondary autoregressive term. In the first and second terms, “a1”, “a2”represent respective gain coefficients of the primary autoregressiveterm and the secondary autoregressive term. Stated otherwise, these gaincoefficients “a1”, “a2” are coefficients relative to the differentialoutput VO2 of the O₂ sensor 6 in the exhaust system model.

The third term of the right side of the equation (1) represents thedifferential output kact of the LAF sensor 5 as an input to the objectexhaust system E, including the dead time d1 of the object exhaustsystem E. In the third term, “b1” represents a gain coefficient relativeto its input (=the differential output kact of the LAF sensor 5). Thegain coefficients “a1”, “a2”, “b1” are parameters which are to be set(identified) to values in defining the behavior of the exhaust systemmodel, and are sequentially identified by an identifier which will bedescribed later on.

In the exhaust system model expressed as the discrete time systemaccording to the equation (1), the differential output VO2(k+1) of theO₂ sensor 6 as an output of the object exhaust system E in each controlcycle of the exhaust-side control unit 7 a is expressed by a pluralityof (two in this embodiment) differential outputs VO2(k), VO(k−1) and adifferential output kact(k−d1) of the LAF sensor 5 in past controlcycles prior to the control cycle.

The model of the air-fuel ratio manipulating system comprising theinternal combustion engine 1 and the engine-side control unit 7 b(hereinafter referred to as an “air-fuel ratio manipulating systemmodel) is expressed as a discrete-time system model according to thefollowing equation (2):

 kact(k)=kcmd(k−d2)  (2)

where “d2” represents the dead time (second dead time) of the air-fuelratio manipulating system in terms of the number of control cycles ofthe exhaust-side control unit 7 a. The dead time of the air-fuel ratiomanipulating system (the time required until the target air-fuel ratioKCMD or the command differential air-fuel ratio kcmd at each point oftime is reflected in the output signal KACT or the differential outputkact of the LAF sensor 5) varies with the rotational speed of theinternal combustion engine 1, and is longer as the rotational speed ofthe internal combustion engine 1 is lower. In present embodiment, inview of the above characteristics of the dead time of the air-fuel ratiomanipulating system expressed by the equation (2), a preset constantvalue (d2=3 in the present embodiment) which is equal to or slightlylonger than the dead time of the actual air-fuel ratio manipulatingsystem at an idling rotational speed of the internal combustion engine 1(the dead time is a maximum dead time which can be taken by the air-fuelratio manipulating system at an arbitrary rotational speed of theinternal combustion engine 1) is used as the value of the dead time d2in the air-fuel ratio manipulating system model expressed by theequation (2).

In the air-fuel ratio manipulating system model expressed by theequation (2), the differential output kact(k) of the LAF sensor 5 ineach control cycle of the exhaust-side control unit 5 a is assumed toagree with the command differential air-fuel ratio kcmd(k−d2) prior tothe dead time d2 of the air-fuel ratio manipulating system, and isexpressed by the command differential air-fuel ratio kcmd(k−d2).

The air-fuel ratio manipulating system actually includes a responsedelay of the internal combustion engine 1 in addition to the dead time.Since a response delay of the output KACT or the differential outputkact of the LAF sensor 5 with respect to the target air-fuel ratio KCMDor the command differential air-fuel ratio kcmd is basically compensatedfor by the feedback controller 14 (particularly the adaptive controller18) in the engine-side control unit 7 b, there will arise no problem ifthe response delay of the internal combustion engine 1 is not taken intoaccount in the air-fuel ratio manipulating system as viewed from themanipulated variable generator 29 in the exhaust-side control unit 7 a.

The manipulated variable generator 29 carries out the processconstructed on the basis of the exhaust system model and the air-fuelratio manipulating system model expressed by the respective equations(1), (2) in the control cycles of the exhaust-side control unit 7 a, tosequentially generate the demand manipulated quantity usl as an input tobe given to the object exhaust system E for converging the outputVO2/OUT of the O₂ sensor 6 to its target value VO2/TARGET. In order togenerate the demand manipulated quantity usl, the manipulated variablegenerator 29 has its functional components as shown in FIG. 3.

Specifically, the manipulated variable generator 29 comprises anidentifier 25 for sequentially identifying in each control cycle valuesof the gain coefficients a1, a2, b1 that are parameters to beestablished for the exhaust system model (the equation (1)), using thedata of the differential output kact from the LAF sensor 5 and thedifferential output VO2 from the O₂ sensor 6, an estimator 26 forsequentially estimating in each control cycle an estimated value VO2 barof the differential output VO2 from the O₂ sensor 6 (hereinafterreferred to as an “estimated differential output VO2 bar”) after thetotal dead time d (=d1+d2) which is the sum of the dead time d1 of theobject exhaust system E and the dead time d2 of the air-fuel ratiomanipulating system, using the data of the differential output kact fromthe LAF sensor 5, the data of the differential output VO2 from the O₂sensor 6, the data of the command differential air-fuel ratio kcmd(normally, kcmd=usl) produced when the demand differential air-fuelratio usl determined in the past by a sliding mode controller 27 islimited by the limiter 30, identified values a1 hat, a2 hat, b1 hat ofthe gain coefficients a1, a2, b1 that are calculated by the identifier25 (hereinafter referred to as “identified gain coefficients a1 hat, a2hat, b1 hat”), and a sliding mode controller 27 for sequentiallydetermining in each control cycle the demand differential air-fuel ratiousl, using the data of the estimated differential output VO2 bar fromthe O₂ sensor 6 which has been determined by the estimator 26 and theidentified gain coefficients a1 hat, a2 hat, b1 hat, according to anadaptive slide mode control process.

The identifier 25 and the estimator 26 correspond respectively to anidentifying means and an estimating means.

The algorithm of a processing operation to be carried out by theidentifier 25, the estimator 26, and the sliding mode controller 27 isconstructed as follows:

The identifier 25 serves to identify the gain coefficients a1, a2, b1sequentially on a real-time basis for the purpose of minimizing amodeling error of the actual object exhaust system E of the exhaustsystem model expressed by the equation (1). The identifier 25 carriesout its identifying process as follows:

In each control cycle of the exhaust-side control unit 7 a, theidentifier 25 determines an identified value VO2(k) hat of thedifferential output VO2 from the O₂ sensor 6 (hereinafter referred to asan “identified differential output VO2 hat”) on the exhaust systemmodel, using the identified gain coefficients a1 hat, a2 hat, b1 hat ofthe presently established exhaust system model, i.e., identified gaincoefficients a1 hat (k−1), a2 hat (k−1), b1 hat (k−1) determined in apreceding control cycle, and past data of the differential output kactfrom the LAF sensor 5 and the differential output VO2 from the O₂ sensor6, according to the following equation (3):

VÔ2(k)=â1(k−1)·VO2(k−1)+â2(k−1)·VO2(k−2)+{circumflex over(b)}1(k−1)·kact(k−d1−1)  (3)

The equation (3) corresponds to the equation (1) which is shifted intothe past by one control cycle with the gain coefficients a1, a2, b1being replaced with the respective identified gain coefficients a1 hat(k−1), a2 hat (k−1), b1 hat (k−1). The value of the dead time “d1” ofthe object exhaust system E in the third term of the equation (3)represents a preset constant value (d1=7 in this embodiment) asdescribed above.

If vectors Θ, ξ defined by the following equations (4), (5) areintroduced (the letter T in the equations (4), (5) and other equationsrepresents a transposition), then the equation (3) is expressed by theequation (6):

Θ^(T)(k)=[â1(k)â2(k){circumflex over (b)}1(k)]  (4)

ξ^(T)(k)=[VO2(k−1)VO2(k−2)kact(k−d1−1)]  (5)

VÔ2(k)=Θ^(T)(k−1)·ξ(k)  (6)

The identifier 25 also determines a difference id/e between theidentified differential output VO2 hat from the O₂ sensor 6 which isdetermined by the equation (3) or (6) and the present differentialoutput VO2 from the O₂ sensor 6, as representing a modeling error of theexhaust system model with respect to the actual object exhaust system E(hereinafter the difference id/e will be referred to as an “identifiederror id/e”), according to the following equation (7):

id/e(k)=VO2(k)−VÔ2(k)  (7)

The identifier 25 further determines new identified gain coefficientsa1(k) hat, a2(k) hat, b1(k) hat, stated otherwise, a new vector Θ(k)having these identified gain coefficients as elements (hereinafter thenew vector Θ(k) will be referred to as an “identified gain coefficientvector Θ”), in order to minimize the identified error id/e, according tothe equation (8) given below. That is, the identifier 25 varies theidentified gain coefficients a1 hat (k−1), a2 hat (k−1), b1 hat (k−1)determined in the preceding control cycle by a quantity proportional tothe identified error id/e for thereby determining the new identifiedgain coefficients a1(k) hat, a2(k) hat, b1(k) hat.

Θ(k)=Θ(k−1)+Kθ(k)·id/e(k)  (8)

where Kθ represents a cubic vector determined by the following equation(9), i.e., a gain coefficient vector for determining a change dependingon the identified error id/e of the identified gain coefficients a1 hat,a2 hat, b1 hat): $\begin{matrix}{{K\quad {\theta (k)}} = \frac{{P( {k - 1} )}\xi \quad (k)}{1 + {{\xi^{T}(k)}{P( {k - 1} )}{\xi (k)}}}} & (9)\end{matrix}$

where P represents a cubic square matrix determined by a recursiveformula expressed by the following equation (10): $\begin{matrix}{{P(k)} = {{\frac{1}{\lambda_{1}(k)}\lbrack {I - \frac{{\lambda_{2}(k)}{P( {k - 1} )}{\xi (k)}{\xi^{T}(k)}}{{\lambda_{1}(k)} + {{\lambda_{2}(k)}{\xi^{T}(k)}{P( {k - 1} )}{\xi (k)}}}} \rbrack}{P( {k - 1} )}}} & (10)\end{matrix}$

where I represents a unit matrix.

In the equation (10), λ₁, λ₂ are established to satisfy the conditions0<λ₁≦1 and 0≦λ₂<2, and an initial value P(0) of P represents a diagonalmatrix whose diagonal components are positive numbers.

Depending on how λ₁, λ₂ in the equation (10) are established, any one ofvarious specific algorithms including a fixed gain method, a degressivegain method, a method of weighted least squares, a method of leastsquares, a fixed tracing method, etc. may be employed. According topresent embodiment, a method of least squares (λ₁=λ₂=1), for example, isemployed.

Basically, the identifier 25 sequentially determines in each controlcycle the identified gain coefficients a1 hat, a2 hat, b1 hat of theexhaust system model in order to minimize the identified error id/eaccording to the above algorithm (calculating operation). Through thisoperation, it is possible to sequentially obtain the identified gaincoefficients a1 hat, a2 hat, b1 hat which match the actual objectexhaust system E.

The calculating operation described above is the basic processing thatis carried out by the identifier 25. In present embodiment, theidentifier 25 performs additional processes such as a limiting process,on the identified gain coefficients a1 hat, a2 hat, b1 hat in order todetermine them. Such operations of the identifier 25 will be describedlater on.

The estimator 26 sequentially determines in each control cycle theestimated differential output VO2 bar which is an estimated value of thedifferential output VO2 from the O₂ sensor 6 after the total dead time d(=d1+d2) in order to compensate for the effect of the dead time d1 ofthe object exhaust system E and the effect of the dead time d2 of theair-fuel ratio manipulating system for the calculation of the demanddifferential air-fuel ratio usl with the sliding mode controller 27 asdescribed in detail later on. Since details of the estimator 26 aredisclosed in U.S. patent application Ser. No. 09/311353, the estimator26 will briefly be described below.

If the equation (2) expressing the model of the air-fuel ratiomanipulating system is applied to the equation (1) expressing theexhaust system model, then the equation (1) can be rewritten as thefollowing equation (11):

VO2(k+2)=a1·VO2(k)+a2·VO2(k−1)+b1·kcmd(k−d1−d2)=a1·VO2(k)+a2·VO2(k−1)+b1·kcmd(k−d)  (11)

The equation (11) expresses a system which is a combination of theobject exhaust system E and the air-fuel ratio manipulating system asthe model of a discrete time system, regarding such a system as a systemfor generating the differential output VO2 of the O₂ sensor 6 from thecommand differential air-fuel ratio kcmd with dead times of the objectexhaust system E and the air-fuel ratio manipulating system and aresponse delay of the object exhaust system E.

Using the equation (11), the estimated differential output VO2(k+d) barwhich is an estimated value of the differential output VO2 of the O₂sensor 6 after the total dead time d in each control cycle is expressedusing present and past time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6 and past time-series datakcmd(k−j) (j=1, 2, . . . , d) of the command differential air-fuel ratiokcmd (=KCMD−FLAF/BASE), according to the following equation (12):$\begin{matrix}\begin{matrix}{{\overset{\_}{VO2}( {k + d} )} = \quad {{{\alpha 1} \cdot {{VO2}(k)}} + {{\alpha 2} \cdot {{VO2}( {k - 1} )}} +}} \\{\quad {\sum\limits_{j = 1}^{d}{\beta_{j} \cdot {{kcmd}( {k - j} )}}}}\end{matrix} & (12)\end{matrix}$

where

α1=the first-row, first-column element of A^(d),

α2=the first-row, second-column element of A^(d),

βj=the first-row elements of A^(j−1)·B $A = \begin{bmatrix}{a1} & {a2} \\1 & 0\end{bmatrix}$ $B = \begin{bmatrix}{b1} \\0\end{bmatrix}$

In the equation (12), α1, α2 represent the first-row, first-columnelement and the first-row, second-column element of the dth power A^(d)(d: total dead time) of the matrix A defined as described above, and βjrepresents the first-row elements of the product A^(j−1)·B of the(j−1)th power A^(j−1) (j=1, 2, . . . , d) of the matrix A and the vectorB defined as described above.

Of the past time-series data kcmd(k−j) (j=1, 2, . . . , d) of thecommand differential air-fuel ratio kcmd in the equation (12), the pasttime-series data kcmd(k−d2), kcmd(k−d2−1), . . . , kcmd(k−d) of thecommand differential air-fuel ratio kcmd before the dead time d2 of theair-fuel ratio manipulating system from the present time can be replacedwith data kact(k), kact(k−1), . . . , kact(k−d+2), respectively, priorto the present time, of the differential output kact of the LAF sensor5. The replacement of the data results in the following equation (13):$\begin{matrix}\begin{matrix}{{\overset{\_}{VO2}( {k + d} )} = \quad {{{\alpha 1} \cdot {{VO2}(k)}} + {{\alpha 2} \cdot {{VO2}( {k - 1} )}} +}} \\{\quad {{\sum\limits_{j = 1}^{d_{2} - 1}{\beta_{j} \cdot {{kcmd}( {k - j} )}}} + {\sum\limits_{i = 0}^{d - d_{2}}{\beta_{i + {d2}} \cdot {{kact}( {k - i} )}}}}} \\{= \quad {{{\alpha 1} \cdot {{VO2}(k)}} + {{\alpha 2} \cdot {{VO2}( {k - 1} )}} +}} \\{\quad {{\sum\limits_{j = 1}^{d_{2} - 1}{\beta_{j} \cdot {{kcmd}( {k - j} )}}} + {\sum\limits_{i = 0}^{d_{1}}{\beta_{i + {d2}} \cdot {{kact}( {k - i} )}}}}}\end{matrix} & (13)\end{matrix}$

The above equation (13) is a basic equation for the estimator 26 tocalculate the estimated differential output VO2(k+d) bar in thisembodiment. Stated otherwise, the estimator 26 determines the estimateddifferential output VO2 bar of the O₂ sensor 6 according to the equation(13), using the present and past time-series data VO2(k), VO2(k−1) ofthe differential output VO2 of the O₂ sensor 6, the past data kcmd(k−j)(j=1, . . . , d2−1) of the command differential air-fuel ratio kcmdwhich is produced by limiting the demand differential air-fuel ratio uslgenerated by the sliding mode controller 27, and the present and pasttime-series data kact(k−i) (i=0, . . . , d1) of the differential outputkact of the LAF sensor 5.

In this embodiment, the values of the coefficients α1, α2, βj requiredto calculate the estimated differential output VO2(k+d) bar according tothe equation (13) are basically calculated using the identified gaincoefficients a1 hat, a2 hat, b1 hat which are identified values of thegain coefficients a1, a2, b1 (these are elements of the matrix A and thevector B defined above with respect to the equation (12)). For thevalues of the dead times d1, d2 required for the calculation of theequation (13), the preset values described above are used.

The estimated differential output VO2(k+d) bar may be determinedaccording to the equation (12) without using the data of thedifferential output kact of the LAF sensor 5. In such a case, theestimated differential output VO2(k+d) bar is determined using thepresent and past time-series data VO2(k), VO2(k−1) of the differentialoutput VO2 of the O₂ sensor 6, the past time-series data kcmd(k−j) (j=1,2, . . . , d) of the command differential air-fuel ratio kcmd, and thevalues of the coefficients α1, α2, βj (j=1, 2, . . . , d) determined bythe identified gain coefficients a1 hat, a2 hat, b1 hat. For increasingthe reliability of the estimated differential output VO2(k+d) bar, it ispreferable to calculate the estimated differential output VO2(k+d) baraccording to the equation (13) using the data of the differential outputkact of the LAF sensor 5 which reflects the actual behavior of theinternal combustion engine 1.

If the dead time d2 of the air-fuel ratio manipulating system may be setto d2=1, then all the past time-series data kcmd(k−j) (j=1, 2, . . . ,d) of the command differential air-fuel ratio kcmd in the equation (12)may be replaced with the time-series data kact(k), kact(k−1), . . . ,kact(k−d+d2), prior to the present time, of the LAF sensor 5. In thiscase, the estimated differential output VO2(k+d) bar can be determinedaccording to the following equation (14) which includes no data of thecommand differential air-fuel ratio kcmd: $\begin{matrix}\begin{matrix}{{\overset{\_}{VO2}( {k + d} )} = \quad {{{\alpha 1} \cdot {{VO2}(k)}} + {{\alpha 2} \cdot {{VO2}( {k - 1} )}} +}} \\{\quad {\sum\limits_{j = 0}^{d - 1}{\beta_{j + 1} \cdot {{kact}( {k - j} )}}}}\end{matrix} & (14)\end{matrix}$

In the present embodiment, the estimator 26 determines in each controlcycle the estimated differential output VO2 bar of the O₂ sensoraccording to the equation (13) which differs from the equation (12) inthat all, prior to the dead time d2 of the air-fuel ratio manipulatingsystem, of the time-series data of the target differential air-fuelratio kcmd in the equation (12) are replaced with the differentialoutput kact of the LAF sensor 5. However, the estimated differentialoutput VO2 bar may be determined according to an equation which differsfrom the equation (12) in that only part of the time-series data of thetarget differential air-fuel ratio kcmd prior to the dead time d2 in theequation (12) is replaced with the differential output kact of the LAFsensor 5.

The above calculating process is the basic algorithm for the estimator26 to determine the estimated differential output VO2(k+d) bar which isan estimated value after the total dead time d of the differentialoutput VO2 of the O₂ sensor in each control cycle.

The sliding mode controller 27 will be described in detail below. Sincedetails of the sliding mode controller 27 are disclosed in U.S. patentapplication Ser. No. 09/311353, the sliding mode controller 27 willbriefly be described below.

The sliding mode controller 27 sequentially determines the demanddifferential air-fuel ratio usl as a manipulated variable formanipulating the air-fuel ratio of the internal combustion engine 1 inorder to converge the output signal VO2/OUT from the O₂ sensor 6 to thetarget value VO2/TARGET, i.e., to converge the differential output VO2of the O₂ sensor 6 to “0”, according to an adaptive sliding mode controlprocess which incorporates an adaptive control law for minimizing theeffect of a disturbance, in the normal sliding mode control process. Analgorithm for carrying out the adaptive sliding mode control process isconstructed as follows:

A switching function required for the adaptive sliding mode controlprocess of the sliding mode controller 27 and a hyperplane defined bythe switching function (also referred to as a slip plane) will first bedescribed below.

According to a basic concept of the sliding mode control process in thepresent embodiment, the differential output VO2(k) from the O₂ sensor 6in each control cycle and the differential output VO2(k−1) in eachpreceding control cycle are used, and a switching function σ for thesliding mode control process is established according to the followingequation (15). The switching function σ is defined by a linear functionhaving as components the present and past time-series data VO2(k),VO2(k−1) of the differential output VO2 of the O₂ sensor 6. The vector Xdefined according to the equation (15) as a vector having thedifferential outputs VO2(k), VO2(k−1) as its components will hereinafterbe referred to as a state quantity X. $\begin{matrix}{\begin{matrix}{{\sigma (k)} = {{{s1} \cdot {{VO2}(k)}} + {{s2} \cdot {{VO2}( {k - 1} )}}}} \\{= {S \cdot X}}\end{matrix}( {{S = \begin{bmatrix}{s1} & {s2}\end{bmatrix}},{X = \begin{bmatrix}{{VO2}(k)} \\{{VO2}( {k - 1} )}\end{bmatrix}}} )} & (15)\end{matrix}$

The coefficients s1, s2 relative to the components VO2(k), VO2(k−1) ofthe switching function σ are established to meet the condition of thefollowing equation (16): $\begin{matrix}{{{- 1} < \frac{s2}{s1} < 1}( {{{{when}{\quad \quad}{s1}} = 1},\quad {{- 1} < {s2} < 1}} )} & (16)\end{matrix}$

In the present embodiment, for the sake of brevity, the coefficient s1is set to s1=1 (s2/s1=s2), and the coefficient s2 is established tosatisfy the condition: −1<s2<1.

With the switching function σ thus defined, the hyperplane for thesliding mode control process is defined by the equation σ=0. Since thestate quantity X is of the second degree, the hyperplane σ=0 isrepresented by a straight line as shown in FIG. 4, and, at this time,the hyperplane is called also a switching function.

The time-series data of the estimated differential output VO2 bardetermined by the estimator 26 is actually used as the components of theswitching function, as described later on.

The adaptive sliding mode control process in this embodiment serves toconverge the state quantity X onto the hyperplane σ=0 according to areaching control law which is a control law for converging the statequantity X=(VO2(k), VO2(k−1)) onto the hyperplane σ=0, i.e., forconverging the value of the switching function σ to “0”, and an adaptivecontrol law (adaptive algorithm) which is a control law for compensatingfor the effect of a disturbance in converging the state quantity X ontothe hyperplane σ=0 (mode 1 in FIG. 4). While converging the statequantity X onto the hyperplane σ=0 according to an equivalent controlinput (holding the value of the switching function σ at “0”), the statequantity X is converted to a balanced point on the hyperplane σ=0 whereVO2(k)=VO2(k−1)=0, i.e., a point where time-series data VO2/OUT(k),VO2/OUT(k−1) of the output VO2/OUT of the O₂ sensor 6 are equal to thetarget value VO2/TARGET.

The command differential air-fuel ratio usl to be generated by thesliding mode controller 27 according to the sliding mode control processfor converging the state quantity X to the balanced point on thehyperplane σ=0 is expressed as the sum of an equivalent control inputueq to be applied to the object exhaust system E according to thecontrol law for converging the state quantity X onto the hyperplane σ=0,an input urch (hereinafter referred to as a “reaching control law inputurch”) to be applied to the object exhaust system E according to thereaching control law, and an input uadp (hereinafter referred to as an“adaptive control law input uadp”) to be applied to the object exhaustsystem E according to the adaptive control law (see the followingequation (17)).

Usl=Ueq+Urch+Uadp  (17)

The equivalent control input ueq, the reaching control law input urch,and the adaptive control law uadp are determined on the basis of themodel of the discrete time system expressed by the equation (11) (amodel where the differential output kact(k−d1) of the LAF sensor 5 inthe equation (1) is replaced with the command differential air-fuelratio kcmd(k−d) using the total dead time d), as follows:

The equivalent control input ueq which is an input to be applied to theobject exhaust system E for converging the state quantity X onto thehyperplane σ=0 is equal to the command differential air-fuel ratio kcmdwhich satisfies the condition: σ(k+1)=σ(k)=0. Using the equations (11),(15), the equivalent control input ueq which satisfies the abovecondition is given by the following equation (18): $\begin{matrix}\begin{matrix}{{{Ueq}(k)} = \quad {{- ( {S \cdot B} )^{- 1}} \cdot \{ {S \cdot ( {A - 1} )} \} \cdot {X( {k + d} )}}} \\{= \quad {\frac{- 1}{s1b1} \cdot \{ {{\lbrack {{{s1} \cdot ( {{a1} - 1} )} + {s2}} \rbrack \cdot {{VO2}( {k + d} )}} +} }} \\{\quad  {( {{{s1} \cdot {a2}} - {s2}} ) \cdot {{VO2}( {k + d - 1} )}} \}}\end{matrix} & (18)\end{matrix}$

The equation (18) is a basic formula for determining the equivalentcontrol input ueq in each control cycle.

According to present embodiment, the reaching control law input urch isbasically determined according to the following equation (19):$\begin{matrix}\begin{matrix}{{{Urch}(k)} = \quad {{- ( {S \cdot B} )^{- 1}} \cdot F \cdot {\sigma ( {k + d} )}}} \\{= \quad {\frac{- 1}{s1b1} \cdot F \cdot {\sigma ( {k + d} )}}}\end{matrix} & (19)\end{matrix}$

Specifically, the reaching control law input urch is determined inproportion to the value σ(k+d) of the switching function σ after thetotal dead time d, in view of the effect of the total dead time d.

The coefficient F in the equation (19) which determines the gain of thereaching control law is established to satisfy the condition expressedby the following equation (20):

0<F<2  (20)

The value of the switching function σ may possibly vary in anoscillating fashion (so-called chattering) with respect to “0”. In orderto suppress such chattering, it is preferable that the coefficient Frelative to the reaching control law input urch be established tofurther satisfy the condition of the following equation (21):

0<F<1  (21)

The adaptive control law input uadp is basically determined according tothe following equation (22) (ΔT in the equation (22) represents theperiod of the control cycles of the exhaust-side control unit 7 a):$\begin{matrix}\begin{matrix}{{{Uadp}(k)} = {{- ( {S \cdot B} )^{- 1}} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}( {{{\sigma (i)} \cdot \Delta}\quad T} )}}} \\{= {\frac{- 1}{s1b1} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}( {{{\sigma (i)} \cdot \Delta}\quad T} )}}}\end{matrix} & (22)\end{matrix}$

The adaptive control law input uadp is determined in proportion to anintegrated value (which corresponds to an integral of the values of theswitching function σ) over control cycles of values of the switchingfunction σ until after the total dead time d, in view of the effect ofthe total dead time d.

The coefficient G (which determines the gain of the adaptive controllaw) in the equation (22) is established to satisfy the condition of thefollowing equation (23): $\begin{matrix}{{G = J}{\cdot \frac{2 - F}{\Delta \quad T}}( {0 < J < 2} )} & (23)\end{matrix}$

A specific process of deriving conditions for establishing the equations(16), (20), (21), (23) is described in detail in Japanese patentapplication No. 11-93741 and U.S. Pat. No. 6,082,099, and will not bedescribed in detail below.

The demand differential air-fuel ratio usl as an input to be given tothe object exhaust system E for converging the output signal VO2/OUT ofthe O₂ sensor 6 to its target value VO2/TARGET may basically bedetermined as the sum (ueq+urch+uadp) of the equivalent control inputueq, the reaching control law input urch, and the adaptive control lawuadp determined according to the respective equations (18), (19), (22).However, the differential outputs VO2(K+d), VO2(k+d−1) of the O₂ sensor6 and the value σ(k+d) of the switching function σ, etc. used in theequations (18), (19), (22) cannot directly be obtained as they arevalues in the future.

According to present embodiment, therefore, the sliding mode controller27 uses the estimated differential outputs VO2(k+d) bar, VO2(k+d−1) bardetermined by the estimator 26, instead of the differential outputsVO2(K+d), VO2(k+d−1) from the O2 sensor 6 for determining the equivalentcontrol input ueq according to the equation (18), and calculates theequivalent control input ueq in each control cycle according to thefollowing equation (24): $\begin{matrix}\begin{matrix}{{{Ueq}(k)} = \quad {\frac{- 1}{s1b1}\{ {{\lbrack {{{s1} \cdot ( {{a1} - 1} )} + {s2}} \rbrack \cdot {\overset{\_}{VO2}( {k + d} )}} +} }} \\ \quad {( {{{s1} \cdot {a2}} - {s2}} ) \cdot {\overset{\_}{VO2}( {k + d - 1} )}} \}\end{matrix} & (24)\end{matrix}$

According to present embodiment, furthermore, the sliding modecontroller 27 actually uses time-series data of the estimateddifferential output VO2 bar sequentially determined by the estimator 26as described as a state quantity to be controlled. That is, the slidingmode controller 27 defines a switching function G bar according to thefollowing equation (25) (the linear function σ bar corresponds totime-series data of the differential output VO2 in the equation (15)which is replaced with time-series data of the estimated differentialoutput VO2 bar), in place of the switching function σ establishedaccording to the equation $\begin{matrix}{{(15)\text{:}}\quad} & \quad \\{\overset{\_}{\sigma (k)} = {{{s1} \cdot {\overset{\_}{VO2}(k)}} + {{s2} \cdot {\overset{\_}{VO2}( {k - 1} )}}}} & (25)\end{matrix}$

The sliding mode controller 27 calculates the reaching control law inputurch in each control cycle according to the following equation (26),using the value of the switching function σ bar represented by theequation (25), rather than the value of the switching function σ fordetermining the reaching control law input urch according to theequation (19): $\begin{matrix}{{{Urch}(k)} = {\frac{- 1}{s1b1} \cdot F \cdot {\overset{\_}{\sigma}( {k + d} )}}} & (26)\end{matrix}$

Similarly, the sliding mode controller 27 calculates the adaptivecontrol law input uadp in each control cycle according to the followingequation (27), using the value of the switching function σ barrepresented by the equation (25), rather than the value of the switchingfunction σ for determining the adaptive control law input uadp accordingto the equation (22): $\begin{matrix}{{{Uadp}(k)} = {\frac{- 1}{s1b1} \cdot G \cdot {\sum\limits_{i = 0}^{k + d}( {{{\overset{\_}{\sigma}(i)} \cdot \Delta}\quad T} )}}} & (27)\end{matrix}$

The latest identified gain coefficients a1(k) hat, a2(k) hat, b1(k) hatwhich have been determined by the identifier 25 are basically used asthe gain coefficients a1, a2, b1 that are required to calculate theequivalent control input ueq, the reaching control law input urch, andthe adaptive control law input uadp according to the equations (24),(26), (27).

The sliding mode controller 27 determines the sum of the equivalentcontrol input ueq, the reaching control law input urch, and the adaptivecontrol law input uadp determined according to the equations (24), (26),(27), as the demand differential air-fuel ratio usl (see the equation(17)). The conditions for establishing the coefficients s1, s2, F, Gused in the equations (24), (26), (27) are as described above.

The demand differential air-fuel ratio usl determined by the slidingmode controller 27 as described above signifies an input to be given tothe object exhaust system 1 for converting the estimated differentialoutput VO2 bar from the O₂ sensor 6 to “0”, and as a result, forconverting the output VO2/OUT from the O₂ sensor 6 to the target valueVO2/TARGET, i.e., a target value for the difference between the air-fuelratio of the internal combustion engine 1 and the reference valueFLAF/BASE.

The above process is a calculating process (algorithm) for generatingthe demand differential air-fuel ratio usl in each control cycle by thesliding mode controller 27.

As described above, the limiter 30 (see FIG. 3) in the target air-fuelratio generator 28 of the exhaustside control unit 7 a limits the demanddifferential air-fuel ratio usl generated by the sliding mode controller27 thereby to determine the command differential air-fuel ratio kcmd(normally kcmd=usl). Then, the adder 31 adds the air-fuel ratioreference value FLAF/BASE to the determined command differentialair-fuel ratio kcmd for thereby determining the target air-fuel ratioKCMD (=kcmd+FLAF/BASE) to be given to the engine-side control unit 7 baccording to the following equation (28):

KCMD(k)=kcmd(k)+FLAF/BASE  (28)

In this embodiment, the limiter 30 sequentially variably sets up anallowable range for limiting the demand differential air-fuel ratio usldepending on the demand differential air-fuel ratio usl and theoperating state of the internal combustion engine 1, as described lateron.

In the present embodiment, furthermore, the air-fuel ratio referencevalue FLAF/BASE used as a reference for the demand differential air-fuelratio usl and the command differential air-fuel ratio kcmd issequentially variably established by the reference value setting unit 11depending on the adaptive control law uadp which is an adaptive controllaw component of the demand differential air-fuel ratio usl generated bythe sliding mode controller 27, as described later on.

The general feedback controller 15 of the engine-side control unit 7 b,particularly, the adaptive controller 18, will be described below.

As shown in FIG. 1, the general feedback controller 15 effects afeedback control process to converge the output KACT (detected air-fuelratio) from the LAF sensor 5 to the target air-fuel ratio KCMD asdescribed above. If such a feedback control process were carried outunder the known PID control only, it would be difficult keep stablecontrollability against dynamic behavioral changes including changes inthe operating conditions of the internal combustion engine 1,characteristic changes due to aging of the internal combustion engine 1,etc.

The adaptive controller 18 is a recursive-type controller which makes itpossible to carry out a feedback control process while compensating fordynamic behavioral changes of the internal combustion engine 1. As shownin FIG. 5, the adaptive controller 18 comprises a parameter adjuster 32for establishing a plurality of adaptive parameters using the parameteradjusting law proposed by I. D. Landau, et al., and a manipulatedvariable calculator 33 for calculating the feedback manipulated variableKSTR using the established adaptive parameters.

The parameter adjuster 32 will be described below. According to theparameter adjusting law proposed by I. D. Landau, et al., whenpolynomials of the denominator and the numerator of a transfer functionB(Z⁻¹)/A(Z⁻¹) of a discrete-system object to be controlled are generallyexpressed respectively by equations (29), (30), given below, an adaptiveparameter θ hat (j) (j indicates the number of a control cycle)established by the parameter adjuster 32 is represented by a vector(transposed vector) according to the equation (31) given below. An inputζ(j) to the parameter adjuster 32 is expressed by the equation (32)given below. In the present embodiment, it is assumed that the internalcombustion engine 1, which is an object to be controlled by the generalfeedback controller 15, is considered to be a plant of a first-ordersystem having a dead time d_(p) corresponding to three control cycles (atime corresponding to three combustion cycles of the internal combustionengine 1), and m=n=1, d_(p)=3 in the equations (29)-(32), and fiveadaptive parameters s₀, r₁, r₂, r₃, b₀ are established (see FIG. 5). Inthe upper and middle expressions of the equation (32), u_(s), y_(s)generally represent an input (manipulated variable) to the object to becontrolled and an output (controlled variable) from the object to becontrolled. In the present embodiment, the input is the feedbackmanipulated variable KSTR and the output from the object (the internalcombustion engine 1) is the output KACT (detected air-fuel ratio) fromthe LAF sensor 5, and the input ζ(j) to the parameter adjuster 32 isexpressed by the lower expression of the equation (32) (see FIG. 5).

A(Z⁻¹)=1+a1Z⁻¹+ . . . +anZ^(−n)  (29)

B(Z⁻¹)=b0+b1Z⁻¹+ . . . +bmZ^(−m)  (30)

 {circumflex over (θ)}^(T)(j)=[{circumflex over (b)}₀(j),{circumflexover (B)}_(R)(Z⁻¹,j),Ŝ(Z⁻¹,j)]=[b₀(j),r₁(j), . . . , r_(m+d) _(p)⁻¹(j),s₀(j), . . . , s_(n−1)(j)]=[b₀(j),r₁(j),r₂(j),r₃(j),s₀(j)]  (31)

ζ^(T)(j)=[u_(s)(j), . . . , u_(s)(j−m−dp+1),y_(s)(j), . . . ,y_(s)(j−n+1)]=[u_(s)(j),u_(s)(j−1),u_(s)(j−2),u_(s)(j−3),y_(s)(j)]=[KSTR(j),KSTR(j−1),KSTR(j−2),KSTR(j−3),KACT(j)]  (32)

The adaptive parameter θ hat expressed by the equation (31) is made upof a scalar quantity element b₀ hat (Z⁻¹,j) for determining the gain ofthe adaptive controller 18, a control element B_(R) hat (Z⁻¹,j)expressed using a manipulated variable, and a control element S hat(Z⁻¹,j) expressed using a controlled variable, which are expressedrespectively by the following equations (33)˜(35) (see the block of themanipulated variable calculator 33 shown in FIG. 5): $\begin{matrix}{{{\hat{b}}_{0}^{- 1}(j)} = \frac{1}{b_{\rho}}} & (33)\end{matrix}$

 {circumflex over (B)}_(R)(Z⁻¹j)=r₁Z⁻¹+r₂Z⁻²+ . . . +r_(m+d) _(p)⁻¹Z^(−(n+dp−1))=r₁Z⁻¹+r₂Z⁻²+r₃Z⁻³  (34)

Ŝ(Z⁻¹,j)=s₀+s₁Z⁻¹+ . . . +s_(n−1)Z^((n−1))=s₀  (35)

The parameter adjuster 32 establishes coefficients of the scalarquantity element and the control elements, described above, and suppliesthem as the adaptive parameter θ hat expressed by the equation (31) tothe manipulated variable calculator 33. The parameter adjuster 32calculates the adaptive parameter θ hat so that the output KACT from theLAF sensor 5 will agree with the target air-fuel ratio KCMD, usingtime-series data of the feedback manipulated variable KSTR from thepresent to the past and the output KACT.

Specifically, the parameter adjuster 32 calculates the adaptiveparameter θ hat according to the following equation (36):

{circumflex over (θ)}(j)={circumflex over(θ)}(j−1)+Γ(j−1)·ζ(j−d_(p))·e*(j)  (36)

where Γ(j) represents a gain matrix (whose degree is indicated bym+n+d_(p)) for determining a rate of establishing the adaptive parameterθ hat, and e*(j) an estimated error of the adaptive parameter θ hat.Γ(j) and e*(j) are expressed respectively by the following recursiveformulas (37), (38): $\begin{matrix}{{\Gamma (j)} = {\frac{1}{\lambda_{1}(j)}\lbrack {{\Gamma ( {j - i} )} - \frac{{\lambda_{2}(j)} \cdot {\Gamma ( {j - 1} )} \cdot {\zeta ( {j - d_{p}} )} \cdot {\zeta^{T}( {j - d_{p}} )} \cdot {\Gamma ( {j - 1} )}}{{\lambda_{1}(j)} + {{\lambda_{2}(j)} \cdot {\zeta^{T}( {j - d_{p}} )} \cdot {\Gamma ( {j - 1} )} \cdot {\zeta ( {j - d_{p}} )}}}} \rbrack}} & (37)\end{matrix}$

where 0<λ₁(j)≦1, 0≦λ₂(j)<2, Γ(0)>0. $\begin{matrix}{{e^{*}(j)} = \frac{{{D( Z^{- 1} )} \cdot {{KACT}(j)}} - {{{\hat{\theta}}^{T}( {j - 1} )} \cdot {\zeta ( {j - d_{p}} )}}}{1 + {{\zeta^{T}( {j - d_{p}} )} \cdot {\Gamma ( {j - 1} )} \cdot {\zeta ( {j - d_{p}} )}}}} & (38)\end{matrix}$

where D(Z⁻¹) represents an asymptotically stable polynomial foradjusting the convergence. In the present embodiment, D(Z⁻¹)=1.

Various specific algorithms including the degressive gain algorithm, thevariable gain algorithm, the fixed trace algorithm, and the fixed gainalgorithm are obtained depending on how λ₁(j) λ₂(j) in the equation (37)are selected. For a time-dependent plant such as a fuel injectionprocess, an air-fuel ratio, or the like of the internal combustionengine 1, either one of the degressive gain algorithm, the variable gainalgorithm, the fixed gain algorithm, and the fixed trace algorithm issuitable.

Using the adaptive parameter θ hat (s₀, r₁, r₂, r₃, b₀) established bythe parameter adjuster 32 and the target air-fuel ratio KCMD calculatedby the exhaust-side control unit 7 a, the manipulated variablecalculator 33 determines the feedback manipulated variable KSTRaccording to a recursive formula expressed by the following equation(39): $\begin{matrix}{{KSTR} = \frac{\begin{matrix}{{{KCMD}(j)} - {s_{0} \cdot {{KACT}(j)}} - {r_{1} \cdot {{KSTR}( {j - 1} )}} -} \\{{r_{2} \cdot {{KSTR}( {j - 2} )}} - {r_{3} \cdot {{KSTR}( {j - 3} )}}}\end{matrix}}{b_{0}}} & (39)\end{matrix}$

The manipulated variable calculator 33 shown in FIG. 5 represents ablock diagram of the calculations according to the equation (39).

The feedback manipulated variable KSTR determined according to theequation (39) becomes the target air-fuel ratio KCMD insofar as theoutput KACT of the LAF sensor 5 agrees with the target air-fuel ratioKCMD. Therefore, the feedback manipulated variable KSTR is divided bythe target air-fuel ratio KCMD by the divider 19 for thereby determiningthe feedback manipulated variable kstr that can be used as the feedbackcorrection coefficient KFB.

As is apparent from the foregoing description, the adaptive controller18 thus constructed is a recursivetype controller taking into accountdynamic behavioral changes of the internal combustion engine 1 which isan object to be controlled. Stated otherwise, the adaptive controller 18is a controller described in a recursive form to compensate for dynamicbehavioral changes of the internal combustion engine 1, and moreparticularly a controller having a recursive-type adaptive parameteradjusting mechanism.

A recursive-type controller of this type may be constructed using anoptimum regulator. In such a case, however, it generally has noparameter adjusting mechanism. The adaptive controller 18 constructed asdescribed above is suitable for compensating for dynamic behavioralchanges of the internal combustion engine 1.

The details of the adaptive controller 18 have been described above.

The PID controller 17, which is provided together with the adaptivecontroller 18 in the general feedback controller 20, calculates aproportional term (P term), an integral term (I term), and a derivativeterm (D term) from the difference between the output KACT of the LAFsensor 5 and the target air-fuel ratio KCMD, and calculates the total ofthose terms as the feedback manipulated variable KLAF, as is the casewith the general PID control process. In the present embodiment, thefeedback manipulated variable KLAF is set to “1” when the output KACT ofthe LAF sensor 5 agrees with the target air-fuel ratio KCMD by settingan initial value of the integral term (I term) to “1”, so that thefeedback manipulated variable KLAF can be used as the feedbackcorrection coefficient KFB for directly correcting the fuel injectionquantity. The gains of the proportional term, the integral term, and thederivative term are determined from the rotational speed and intakepressure of the internal combustion engine 1 using a predetermined map.

The switcher 20 of the general feedback controller 15 outputs thefeedback manipulated variable KLAF determined by the PID controller 17as the feedback correction coefficient KFB for correcting the fuelinjection quantity if the combustion in the internal combustion engine 1tends to be unstable as when the temperature of the coolant of theinternal combustion engine 1 is low, the internal combustion engine 1rotates at high speeds, or the intake pressure is low, or if the outputKACT of the LAF sensor 5 is not reliable due to a response delay of theLAF sensor 5 as when the target air-fuel ratio KCMD changes largely orimmediately after the air-fuel ratio feedback control process hasstarted, or if the internal combustion engine 1 operates highly stablyas when it is idling and hence no high-gain control process by theadaptive controller 18 is required. Otherwise, the switcher 20 outputsthe feedback manipulated variable kstr which is produced by dividing thefeedback manipulated variable KSTR determined by the adaptive controller18 by the target air-fuel ration KCMD, as the feedback correctioncoefficient KFB for correcting the fuel injection quantity. This isbecause the adaptive controller 18 effects a high-gain control processand functions to converge the output KACT of the LAF sensor 5 quickly tothe target air-fuel ratio KCMD, and if the feedback manipulated variableKSTR determined by the adaptive controller 18 is used when thecombustion in the internal combustion engine 1 is unstable or the outputKACT of the LAF sensor 5 is not reliable, then the air-fuel ratiocontrol process tends to be unstable.

Such operation of the switcher 20 is disclosed in detail in Japaneselaid-open patent publication No. 8-105345 and U.S. Pat. No. 5,558,075,and will not be described in detail below.

Operation of the plant control system will be described below.

First, a process, carried out by the engine-side control unit 7 b, ofcalculating an output fuel injection quantity #nTout (n=1, 2, 3, 4) foreach of the cylinders of the internal combustion engine 1 forcontrolling the air-fuel ratio of the internal combustion engine 1 willbe described below with reference to FIG. 6. The engine-side controlunit 7 b calculates an output fuel injection quantity #nTout (n=1, 2, 3,4) for each of the cylinders in synchronism with a crankshaft angleperiod of the internal combustion engine 1 as follows:

In FIG. 7, the engine-side control unit 7 b reads outputs from varioussensors including the LAF sensor 5 and the O₂ sensor 6 in STEPa. At thistime, the output KACT of the LAF sensor 5 and the output VO2/OUT of theO₂ sensor 6, including data obtained in the past, are stored in atime-series fashion in a memory (not shown).

Then, the basic fuel injection quantity calculator 8 corrects a fuelinjection quantity corresponding to the rotational speed NE and intakepressure PB of the internal combustion engine 1 depending on theeffective opening area of the throttle valve, thereby calculating abasic fuel injection quantity Tim in STEPb. The first correctioncoefficient calculator 9 calculates a first correction coefficientKTOTAL depending on the coolant temperature and the amount by which thecanister is purged in STEPc.

The engine control unit 7 b determines an operation mode of the internalcombustion engine 1, and sets a value of a flag f/prism/on whichrepresents whether the target air-fuel ratio KCMD generated by thetarget air-fuel ratio generator 28 of the exhaust-side control unit 7 ais to be used to manipulate the air-fuel ratio of the internalcombustion engine 1 or not in STEPd.

Operation modes of the internal combustion engine 1, specifically modesof manipulating the air-fuel ratio of the internal combustion engine 1,include an operation mode in which the engine-side control unit 7 b usesthe target air-fuel ratio KCMD generated by the target air-fuel ratiogenerator 28 (which is basically a target air-fuel ratio for convergingthe output signal VO2/OUT from the O₂ sensor 6 to the target valueVO2/TARGET), and manipulates the air-fuel ratio of the internalcombustion engine 1 (i.e., adjust the fuel injection quantity of theinternal combustion engine 1) in order to the output signal KACT of theLAF sensor (detected air-fuel ratio) to the target air-fuel ratio KCMD,and an operation mode in which the engine-side control unit 7 bmanipulates the air-fuel ratio of the internal combustion engine 1(i.e., adjust the fuel injection quantity of the internal combustionengine 1), without using the target air-fuel ratio KCMD generated by thetarget air-fuel ratio generator 28.

The former operation mode is a normal operation mode while the internalcombustion engine 1 is in operation (hereinafter referred to as a“normal operation mode”). The latter operation mode represents aplurality of operation modes including an operation mode in which thefuel supplied to the internal combustion engine 1 is cut off (stopped),an operation mode in which the throttle valve (not shown) of theinternal combustion engine 1 is fully open, and an operation mode inwhich the air-fuel ratio of the internal combustion engine 1 ismanipulated to generate a lean air-fuel mixture (lean operation mode).

The deciding process of STEPd is a process of determining whether theinternal combustion engine 1 is in the normal operation mode or not,i.e., whether the engine-side control unit 7 b is to manipulate theair-fuel ratio of the internal combustion engine 1 using the targetair-fuel ratio KCMD generated by the target air-fuel ratio generator 28,or not. When the value of the flag f/prism/on is “0”, it means that thetarget air-fuel ratio KCMD generated by the target air-fuel ratiogenerator 28 is not to be used (the internal combustion engine 1 is notin the normal operation mode), and when the value of the flag f/prism/onis “1”, it means that the target air-fuel ratio KCMD generated by thetarget air-fuel ratio generator 28 is to be used (the internalcombustion engine 1 is in the normal operation mode).

The deciding subroutine of STEPd is shown in detail in FIG. 7. As shownin FIG. 7, the engine-side control unit 7 b decides whether the O₂sensor 6 and the LAF sensor 5 are activated or not respectively inSTEPd-1, STEPd-2. If neither one of the O₂ sensor 6 and the LAF sensor 5is activated, since detected data from the O₂ sensor 6 and the LAFsensor 5 for use by the target air-fuel ratio generator 28 are notaccurate enough, the target air-fuel ratio generator 28 cannot generatean appropriate target air-fuel ratio. Therefore, since the internalcombustion engine 1 cannot operate in the normal operation mode, thevalue of the flag f/prism/on is set to “0” in STEPd-10.

Then, the engine-side control unit 7 b decides whether the internalcombustion engine 1 is operating with a lean air-fuel mixture or not inSTEPd-3. The engine control unit 7 b decides whether the ignition timingof the internal combustion engine 1 is retarded for early activation ofthe catalytic converter 3 immediately after the start of the internalcombustion engine 1 or not in STEPd-4. The engine control unit 7 bdecides whether the throttle valve of the internal combustion engine 1is fully open or not in STEPd-5. The engine control unit 7 b decideswhether the supply of fuel to the internal combustion engine 1 is beingstopped or not in STEPd-6. If either one of the conditions of thesesteps is satisfied (YES), then since the internal combustion engine 1 isnot in the normal operation mode, the value of the flag f/prism/on isset to “0” in STEPd-10.

The engine-side control unit 7 b then decides whether the rotationalspeed NE and the intake pressure PB of the internal combustion engine 1fall within respective given ranges (normal ranges) or not respectivelyin STEPd-7, STEPd-8. If either one of the rotational speed NE and theintake pressure PB does not fall within its given range, then since itis not preferable to control the air-fuel ratio of the internalcombustion engine 1 using the target air-fuel ratio KCMD generated bythe target air-fuel ratio generator 28, the value of the flag f/prism/onis set to “0” in STEPd-10.

If the conditions of STEPd-1, STEPd-2, STEPd-7, STEPd-8 are satisfied,and the conditions of STEPd-3, STEPd-4, STEPd-5, STEPd-6 are notsatisfied (normal), then the engine-side control unit 7 b determinesthat the internal combustion engine 1 is in the normal operation mode inwhich the target air-fuel ratio KCMD generated by the target air-fuelratio generator 28 is used to manipulate the air-fuel ratio of theinternal combustion engine 1, and sets the value of the flag f/prism/onto “1” in STEPd-9.

After the value of the flag f/prism/on is set to “0” in STEPd-10, theengine-side control unit 7 b sets, to a given initial value TMSTB, atimer counter tm/stb (countdown timer) for measuring an elapsed timefrom the time when the value of the flag f/prism/on changes from “0” to“1”, i.e., an elapsed time after the air-fuel ratio of the internalcombustion engine 1 starts being manipulated according to the targetair-fuel ratio KCMD generated by the target air-fuel ratio generator 28,i.e., the control process for converging the output signal VO2/OUT ofthe O₂ sensor 6 to the target value VO2/TARGET has started to be carriedout, and activates the timer counter tm/stb in STEPd-11. The count ofthe timer counter tm/stb is counted down at a constant period (longerthan the period of control cycles of the exhaust-side control unit 7 a).The initial value TMSTB corresponds to a period in which to inhibit thedecision as to the stability of the sliding mode controller 27.

In FIG. 6, after the value of the flag f/prism/on has been set, theengine-side control unit 7 b determines the value of the flag f/prism/onin STEPe. If f/prism/on=1, then the engine-side control unit 7 b readsthe latest target air-fuel ratio KCMD generated by the target air-fuelratio generator 28 in STEPf. If f/prism/on=0, then the engine-sidecontrol unit 7 b sets the target air-fuel ratio KCMD to a predeterminedvalue in STEPg. The predetermined value to be established as the targetair-fuel ratio KCMD is determined from the rotational speed NE andintake pressure PB of the internal combustion engine 1 using apredetermined map, for example. Particularly when the engine 1 is in thelean operation mode, the target air-fuel ratio KCMD set in STEPg is anair-fuel ratio in a lean range.

In the local feedback controller 16, the PID controllers 22 calculatesrespective feedback correction coefficients #nKLAF in order to eliminatevariations between the cylinders, based on actual air-fuel ratios #nA/Fof the respective cylinders which have been estimated from the outputKACT of the LAF sensor 5 by the observer 21, in STEPh. Then, the generalfeedback controller 15 calculates a feedback correction coefficient KFBin STEPi.

Depending on the operating conditions of the internal combustion engine1, the switcher 20 selects either the feedback manipulated variable KLAFdetermined by the PID controller 17 or the feedback manipulated variablekstr which has been produced by dividing the feedback manipulatedvariable KSTR determined by the adaptive controller 18 by the targetair-fuel ratio KCMD (normally, the switcher 20 selects the feedbackmanipulated variable kstr). The switcher 20 then outputs the selectedfeedback manipulated variable KLAF or kstr as a feedback correctioncoefficient KFB.

When switching the feedback correction coefficient KFB from the feedbackmanipulated variable KLAF from the PID controller 17 to the feedbackmanipulated variable kstr from the adaptive controller 18, the adaptivecontroller 18 determines a feedback manipulated variable KSTR in amanner to hold the correction coefficient KFB to the precedingcorrection coefficient KFB (=KLAF) as long as in the cycle time for theswitching. When switching the feedback correction coefficient KFB fromthe feedback manipulated variable kstr from the adaptive controller 18to the feedback manipulated variable KLAF from the PID controller 17,the PID controller 17 calculates a present correction coefficient KLAFin a manner to regard the feedback manipulated variable KLAF determinedby itself in the preceding cycle time as the preceding correctioncoefficient KFB (=kstr).

After the feedback correction coefficient KFB has been calculated, thesecond correction coefficient calculator 10 calculates in STEPj a secondcorrection coefficient KCMDM depending on the target air-fuel ratio KCMDdetermined in STEPf or STEPg.

Then, the engine-side control unit 7 b multiplies the basic fuelinjection quantity Tim, determined as described above, by the firstcorrection coefficient KTOTAL, the second correction coefficient KCMDM,the feedback correction coefficient KFB, and the feedback correctioncoefficients #nKLAF of the respective cylinders, determining output fuelinjection quantities #nTout of the respective cylinders in STEPk. Theoutput fuel injection quantities #nTout are then corrected foraccumulated fuel particles on intake pipe walls of the internalcombustion engine 1 by the fuel accumulation corrector 23 in STEPm. Thecorrected output fuel injection quantities #nTout are applied to thenon-illustrated fuel injectors of the internal combustion engine 1 inSTEPn.

In the internal combustion engine 1, the fuel injectors inject fuel intothe respective cylinders according to the respective output fuelinjection quantities #nTout.

The above calculation of the output fuel injection quantities #nTout andthe fuel injection of the internal combustion engine 1 are carried outin successive cycle times synchronous with the crankshaft angle periodof the internal combustion engine 1 for controlling the air-fuel ratioof the internal combustion engine 1 in order to converge the output KACTof the LAF sensor 5 (the detected air-fuel ratio) to the target air-fuelratio KCMD. While the feedback manipulated variable kstr from theadaptive controller 18 is being used as the feedback correctioncoefficient KFB, the output KACT of the LAF sensor 5 is quicklyconverged to the target air-fuel ratio KCMD with high stability againstbehavioral changes such as changes in the operating conditions of theinternal combustion engine 1 or characteristic changes thereof. Aresponse delay of the internal combustion engine 1 is also appropriatelycompensated for.

Concurrent with the above air-fuel ratio manipulation for the internalcombustion engine 1, the target air-fuel ratio generator 28 of theexhaust-side control unit 7 a executes a main routine shown in FIG. 8 incontrol cycles of a constant period.

As shown in FIG. 8, the target air-fuel ratio generator 28 decideswhether its own processing (the processing of the identifier 25, theestimator 26, the sliding mode controller 27, and the limiter 30) is tobe executed or not, and sets a value of a flag f/prism/cal indicative ofwhether the processing is to be executed or not in STEP1. When the valueof the flag f/prism/cal is “0”, it means that the processing of thetarget air-fuel ratio generator 28 is not to be executed, and when thevalue of the flag f/prism/cal is “1”, it means that the processing ofthe target air-fuel ratio generator 28 is to be executed.

The deciding subroutine in STEP1 is shown in detail in FIG. 9. As shownin FIG. 9, the target air-fuel ratio generator 28 decides whether the O₂sensor 6 and the LAF sensor 5 are activated or not respectively inSTEP1-1, STEP1-2. If neither one of the O₂ sensor 6 and the LAF sensor 5is activated, since detected data from the O₂ sensor 6 and the LAFsensor 5 for use by the target air-fuel ratio generator 28 are notaccurate enough, the value of the flag f/prism/cal is set to “0” inSTEP1-6. Then, in order to initialize the identifier 25 as describedlater on, the value of a flag f/id/reset indicative of whether theidentifier 25 is to be initialized or not is set to “1” in STEP1-7. Whenthe value of the flag f/id/reset is “1”, it means that the identifier 25is to be initialized, and when the value of the flag f/id/reset is “0”,it means that the identifier 25 is not to be initialized.

The target air-fuel ratio generator 28 decides whether the internalcombustion engine 1 is operating with a lean air-fuel mixture or not inSTEP1-3. The target air-fuel ratio generator 28 decides whether theignition timing of the internal combustion engine 1 is retarded forearly activation of the catalytic converter 3 immediately after thestart of the internal combustion engine 1 or not in STEP1-4. If theconditions of these steps are satisfied, then since the target air-fuelratio KCMD calculated to converge the output VO2/OUT of the O₂ sensor 6to the target value VO2/TARGET is not used for the fuel control for theinternal combustion engine 1, the value of the flag f/prism/cal is setto “0” in STEP1-6, and the value of the flag f/id/reset is set to “1” inorder to initialize the identifier 25 in STEP1-7.

If the conditions of STEP1-1, STEP1-2 are satisfied, and the conditionsof STEP1-3, STEP1-4 are not satisfied, then the value of the flagf/prism/cal is set to “1” in STEP1-5.

In FIG. 8, after the above deciding subroutine, the target air-fuelratio generator 28 decides whether a process of identifying (updating)the gain coefficients a1, a1, b1 with the identifier 25 is to beexecuted or not, and sets a value of a flag f/id/cal indicative ofwhether the process of identifying (updating) the gain coefficients a1,a1, b1 is to be executed or not in STEP2.

In STEP2, the target air-fuel ratio generator 28 decides whether thethrottle valve of the internal combustion engine 1 is substantiallyfully open or not, and whether the supply of fuel to the internalcombustion engine 1 is being stopped or not. If either one of theconditions of these steps is satisfied, then since it is difficult toidentify the gain coefficients a1, a1, b1 appropriately, the value ofthe flag f/id/cal is set to “0”. If neither one of the conditions ofthese steps is satisfied, then the value of the flag f/id/cal is set to“1” to identify (update) the gain coefficients a1, a2, b1 with theidentifier 25.

Referring back to FIG. 8, the target air-fuel ratio generator 28calculates the latest differential outputs kact(k) (=KACT−FLAF/BASE),VO2(k) (=VO2/OUT−VO2/TARGET) respectively from the subtractors 12, 13 inSTEP3. Specifically, the subtractors 12, 13 select latest ones of thetime-series data read and stored in the non-illustrated memory in STEPashown in FIG. 6, and calculate the differential outputs kact(k), VO2(k).The air-fuel ratio reference value FLAF/BASE required to calculate thedifferential output kact(k) is of a latest nature set by the referencevalue setting unit 11 as described later on. The data of thedifferential outputs kact(k), VO2(k), as well as the data thereofcalculated in the past, are stored in a time-series manner in a memory(not shown) in the exhaustside control unit 7 a.

Then, in STEP4, the target air-fuel ratio generator 28 determines thevalue of the flag f/prism/cal set in STEP1. If the value of the flagf/prism/cal is “0”, i.e., if the processing of the target air-fuel ratiogenerator 28 is not to be executed, then the target air-fuel ratiogenerator 28 forcibly sets the command differential air-fuel ratio kcmdfor determining the target air-fuel ration KCMD to a predetermined valuein STEP13. The predetermined value may be a fixed value (e.g., “0”) orthe value of the command differential air-fuel ratio kcmd determined ina preceding control cycle. After the command differential air-fuel ratiokcmd is set to the predetermined value in STEP13, the adder 31 adds theair-fuel ratio reference value FLAF/BASE (the latest one set by thereference value setting unit 11) to the command differential air-fuelratio kcmd for thereby determining a target air-fuel ratio KCMD in thepresent control cycle in STEP 11. Then, the reference value setting unit11 carries out a process of setting the air-fuel ratio reference valueFLAF/BASE, as described later on, in STEP12, after which the processingin the present control cycle is finished.

If the value of the flag f/prism/cal is “1” in STEP4, i.e., if theprocessing of the target air-fuel ratio generator 28 is to be executed,then the target air-fuel ratio generator 28 effects the processing ofthe identifier 25 in STEP5.

The processing subroutine of STEP5 is shown in detail in FIG. 10.

The identifier 25 determines the value of the flag f/id/cal set in STEP2in STEP5-1. If the value of the flag f/id/cal is “0” (the throttle valveof the internal combustion engine 1 is substantially fully open or thefuel supply of the internal combustion engine 1 is being cut off), thensince the process of identifying the gain coefficients a1, a2, b1 withthe identifier 25 is not carried out, control immediately goes back tothe main routine shown in FIG. 8.

If the value of the flag f/id/cal is “1”, then the identifier 25determines the value of the flag f/id/reset set in STEP1 with respect tothe initialization of the identifier 25 in STEP5-2. If the value of theflag f/id/reset is “1”, the identifier 25 is initialized in STEP5-3.When the identifier 25 is initialized, the identified gain coefficientsa1 hat, a2 hat, b1 hat are set to predetermined initial values (theidentified gain coefficient vector Θ according to the equation (4) isinitialized), and the elements of the matrix P (diagonal matrix)according to the equation (9) are set to predetermined initial values.The value of the flag f/id/reset is reset to “0”.

Then, the identifier 25 calculates the identified differential outputVO2(k) hat from exhaust system model (see the equation (3)) which isexpressed using the present identified gain coefficients a1(k−1) hat,a2(k−1) hat, b1(k−1) hat, according to the equation (3) or the equation(6) equivalent thereto, using the past data VO2(k−1), VO2(k−2),kact(k−d−1) of the differential outputs VO2, kact calculated in eachcontrol cycle in STEP3, and the identified gain coefficients a1(k−1)hat, a2(k−1) hat, b1(k−1) hat, in STEP5-4.

The identifier 25 then calculates the vector Kθ(k) to be used indetermining the new identified gain coefficients a1 hat, a2 hat, b1 hataccording to the equation (9) in STEP5-5. Thereafter, the identifier 25calculates the identified error id/e, i.e., the difference between theidentified differential output VO2 hat from the O₂ sensor 6 in theexhaust system model and the actual differential output VO2 (see theequation (7)), in STEP5-6.

The identified error id/e obtained in STEP5-6 may basically becalculated according to the equation (7). In the present embodiment,however, a value (=VO2−VO2 hat) calculated according to the equation (7)from the differential output VO2 acquired in each control cycle in STEP3(see FIG. 8), and the identified differential output VO2 hat calculatedin each control cycle in STEP5-4 is filtered with low-passcharacteristics to calculate the identified error id/e.

This is because since the object exhaust system E including thecatalytic converter 3 generally has low-pass characteristics, it ispreferable to attach importance to the low-frequency behavior of theobject exhaust system E in appropriately identifying the gaincoefficients a1, a2, b1 of the exhaust system model.

Both the differential output VO2 and the identified differential outputVO2 hat may be filtered with the same low-pass characteristics. Forexample, after the differential output VO2 and the identifieddifferential output VO2 hat have separately been filtered, the equation(7) may be calculated to determine the identified error idle. The abovefiltering is carried out by a moving average process which is a digitalfiltering process, for example.

After the identifier 25 has calculated the identified error id/e, theidentifier 25 calculates a new identified gain coefficient vector Θ(k),i.e., new identified gain coefficients a1(k) hat, a2(k) hat, b1(k) hat,according to the equation (8) using the identified error id/e and Kθcalculated in SETP5-5 in STEP5-7.

After having calculated the new identified gain coefficients a1(k) hat,a2(k) hat, b1(k) hat, the identifier 25 further limits the values of thegain coefficients a1 hat, a2 hat, b1 hat (elements of the identifiedgain coefficient vector Θ), are limited to meet predetermined conditionsin STEP5-8.

The predetermined conditions for limiting the values of the identifiedgain coefficients a1 hat, a2 hat, b1 hat include a condition(hereinafter referred to as a “first limiting condition”) for limitingcombinations of the values of the identified gain coefficients a1 hat,a2 hat relative to a predetermined combination, and a condition(hereinafter referred to as a “second limiting condition”) for limitingthe value of the identified gain coefficient b1 hat.

Prior to describing the first and second limiting conditions and thespecific processing details of STEP5-8, the reasons for limiting thevalues of the identified gain coefficients a1 hat, a2 hat, b1 hat willbe described below.

The inventors of the present invention have found that if the values ofthe identified gain coefficients a1 hat, a2 hat, b1 hat are notparticularly limited, while the output signal VO2/OUT of the O₂ sensor 6is being stably controlled at the target value VO2/TARGET, there aredeveloped a situation in which the demand differential air-fuel ratiousl determined by the sliding mode controller 27 and the target air-fuelratio KCMD change smoothly with time, and a situation in which thedemand differential air-fuel ratio usl and the target air-fuel ratioKCMD oscillate with time at a high frequency. Neither of thesesituations poses problems in controlling the output VO2/OUT of the O₂sensor 6 at the target value VO2/TARGET. However, the situation in whichthe target air-fuel ratio KCMD oscillates with time at a high frequencyis not preferable in smoothly operating the internal combustion engine1.

A study of the above phenomenon by the inventors has shown that whetherthe demand differential air-fuel ratio usl and the target air-fuel ratioKCMD change smoothly or oscillate at a high frequency is affected by thecombinations of the values of the identified gain coefficients a1 hat,a2 hat identified by the identifier 25 and the value of the identifiedgain coefficient b1 hat.

In the present embodiment, the first and second limiting conditions areestablished appropriately, and the combinations of the values of theidentified gain coefficients a1 hat, a2 hat and the value of theidentified gain coefficient b1 hat are appropriately limited toeliminate the situation in which the target air-fuel ratio KCMDoscillates at a high frequency.

According to the present embodiment, the first and second limitingconditions are established as follows:

With respect to the first limiting condition for limiting the values ofthe identified gain coefficients a1 hat, a2 hat, the study by theinventors indicates that obtaining the demand differential air-fuelratio usl and the target air-fuel ratio KCMD is closely related tocombinations of the coefficient values α1, α2 in the equations (12)-(14)which are determined by the values of the gain coefficients a1, a2,i.e., the coefficient values α1, α2 used for the estimator 26 todetermine the estimated differential output VO2(k+d) bar (thecoefficient values α1, α2 are the first-row, first-column element andthe first-row, second-column element of the matrix A^(d) which is apower of the matrix A defined by the equation (12)).

Specifically, as shown in FIG. 11, when a coordinate plane whosecoordinate components are represented by the coefficient values α1, α2is established, if a point on the coordinate plane which is determinedby a combination of the coefficient values α1, α2 lies in a hatchedrange, which is surrounded by a triangle Q₁Q₂Q₃ (including theboundaries) and will hereinafter be referred to as an “estimatingcoefficient stable range”, then the demand differential air-fuel ratiousl and the target air-fuel ratio KCMD tend to be smooth.

Therefore, the combinations of the values of the gain coefficients a1,a2 identified by the identifier 25, i.e., the combinations of the valuesof the identified gain coefficients a1 hat, a2 hat, should be limitedsuch that the point on the coordinate plane shown in FIG. 11 whichcorresponds to the combination of the coefficient values α1, α2determined by the values of the gain coefficients a1, a2 or the valuesof the identified gain coefficients a1 hat, a2 hat will lie within theestimating coefficient stable range.

In FIG. 11, a triangular range Q₁Q₄Q₃ on the coordinate plane whichcontains the estimating coefficient stable range is a range thatdetermines combinations of the coefficient values α1, α2 which makestheoretically stable a system defined according to the followingequation (40), i.e., a system defined by an equation similar to theequation (12) except that VO2(k), VO2(k−1) on the right side of theequation (12) are replaced respectively with VO2(k) bar, VO2(k−1) bar(VO2(k) bar, VO2(k−1) bar mean respectively an estimated differentialoutput determined in each control cycle by the estimator 26 and anestimated differential output determined in a preceding cycle by theestimator 26). $\begin{matrix}{{\overset{\_}{VO2}( {k + d} )} = {{{\alpha 1} \cdot {\overset{\_}{VO2}(k)}} + {{\alpha 2} \cdot {\overset{\_}{VO2}( {k - 1} )}} + {\sum\limits_{j = 1}^{d}{\beta_{j} \cdot {{kcmd}( {k - j} )}}}}} & (40)\end{matrix}$

The condition for the system defined according to the equation (40) tobe stable is that a pole of the system (which is given by the followingequation (41)) exists in a unit circle on a complex plane:

Pole of the system according to the equation (40) $\begin{matrix}{= \frac{{\alpha 1} \pm \sqrt{{\alpha 1}^{2} + {4 \cdot {\alpha 2}}}}{2}} & (41)\end{matrix}$

The triangular range Q₁Q₄Q₃ shown in FIG. 11 is a range for determiningthe combinations of the coefficient values α1, α2 which satisfy theabove condition. Therefore, the estimating coefficient stable range is arange indicative of those combinations where α1≧0 of the combinations ofthe coefficient values α1, α2 which make stable the system defined bythe equation (40).

Since the coefficient values α1, α2 are determined by a combination ofthe values of the gain coefficients a1, a2, a combination of the valuesof the gain coefficients a1, a2 is determined by a combination of thecoefficient values α1, α2. Therefore, the estimating coefficient stablerange shown in FIG. 11 which determines preferable combinations of thecoefficient values α1, α2 can be converted into a range on a coordinateplane shown in FIG. 12 whose coordinate components are represented bythe gain coefficients a1, a2. Specifically, the estimating coefficientstable range shown in FIG. 11 is converted into a range enclosed by theimaginary lines in FIG. 12, which is a substantially triangular rangehaving an undulating lower side and will hereinafter be referred to asan “identifying coefficient stable range”, on the coordinate plane shownin FIG. 12. Stated otherwise, when a point on the coordinate plane shownin FIG. 12 which is determined by a combination of the values of thegain coefficients a1, a2 resides in the identifying coefficient stablerange, a point on the coordinate plane shown in FIG. 11 whichcorresponds to the combination of the coefficient values α1, α2determined by those values of the gain coefficients a1, a2 resides inthe estimating coefficient stable range.

Consequently, the first limiting condition for limiting the values ofthe identified gain coefficients a1 hat, a2 hat determined by theidentifier 25 should preferably be basically established such that apoint on the coordinate plane shown in FIG. 12 which is determined bythose values of the identified gain coefficients a1 hat, a2 hat residein the identifying coefficient stable range.

However, since a boundary (lower side) of the identifying coefficientstable range indicated by the imaginary lines in FIG. 12 is of a complexundulating shape, a practical process for limiting the point on thecoordinate plane shown in FIG. 12 which is determined by the values ofthe identified gain coefficients a1 hat, a2 hat is liable to be complex.

For this reason, according to the present embodiment, the identifyingcoefficient stable range is substantially approximated by a quadrangularrange Q₅Q₆Q₇Q₈ enclosed by the solid lines in FIG. 12, which hasstraight boundaries and will hereinafter be referred to as an“identifying coefficient limiting range”. As shown in FIG. 12, theidentifying coefficient limiting range is a range enclosed by apolygonal line (including line segments Q₅Q₆ and Q₅Q₈) expressed by afunctional expression |a1|+a2=1, a straight line (including a linesegment Q₆Q₇) expressed by a constant-valued functional expressiona1=A1L (A1L: constant), and a straight line (including a line segmentQ₇Q₈) expressed by a constant-valued functional expression a2=A2L (A2L:constant). The first limiting condition for limiting the values of theidentified gain coefficients a1 hat, a2 hat is established such that thepoint on the coordinate plane shown in FIG. 15 which is determined bythose values of the identified gain coefficients a1 hat, a2 hat lies inthe identifying coefficient limiting range. Although part of the lowerside of the identifying coefficient limiting range deviates from theidentifying coefficient stable range, it has experimentally beenconfirmed that the point determined by the identified gain coefficientsa1 hat, a2 hat determined by the identifier 25 does not actually fall inthe deviating range. Therefore, the deviating range will not pose anypractical problem.

The above identifying coefficient limiting range is given forillustrative purpose only, and may be equal to or may substantiallyapproximate the identifying coefficient stable range, or may be of anyshape insofar as most or all of the identifying coefficient limitingrange belongs to the identifying coefficient stable range. Thus, theidentifying coefficient limiting range may be established in variousconfigurations in view of the ease with which to limit the values of theidentified gain coefficients a1 hat, a2 hat and the practicalcontrollability.

For example, while the boundary of an upper portion of the identifyingcoefficient limiting range is defined by the functional expression|a1|+a2=1 in the illustrated embodiment, combinations of the values ofthe gain coefficients a1, a2 which satisfy this functional expressionare combinations of theoretical stable limits where a pole of the systemdefined by the equation (41) exists on a unit circle on a complex plane.Therefore, the boundary of the upper portion of the identifyingcoefficient limiting range may be determined by a functional expression|a1|+a2=r (r is a value slightly smaller than “1” corresponding to thestable limits, e.g., 0.99) for higher control stability.

The above identifying coefficient stable range shown in FIG. 12 as abasis for the identifying coefficient limiting range is given forillustrative purpose only. The identifying coefficient stable rangewhich corresponds to the estimating coefficient stable range shown inFIG. 11 is affected by the dead time d (more precisely, its set value)and has its shape varied depending on the dead time d, as can be seenfrom the definition of the coefficient values α1, α2 (see the equation(12)). Irrespective of the shape of the identifying coefficient stablerange, the identifying coefficient limiting range may be established, asdescribed above, in a manner to match the shape of the identifyingcoefficient stable range.

In the present embodiment, the second limiting condition for limitingthe value of the gain coefficient b1 identified by the identifier 25,i.e., the value of the identified gain coefficient b1 hat, isestablished as follows:

The inventors have found that the situation in which the time-dependingchange of the target air-fuel ratio KCMD is oscillatory at a highfrequency tends to happen also when the value of the identified gaincoefficient b1 hat is excessively large or small. According to thepresent embodiment, an upper limit value B1H and a lower limit value B1L(B1H>B1L>0) for the identified gain coefficient b1 hat are determined inadvance through experimentation or simulation. Then, the second limitingcondition is established such that the identified gain coefficient b1hat is equal to or smaller than the upper limit value B1H and equal toor greater than the lower limit value B1L (B1L≦b1 hat≦B1H).

A process of limiting the values of the identified gain coefficients a1hat, a2 hat, b1 hat according to the first and second limitingconditions is carried out by in STEP5-8 as follows:

As shown in FIG. 13, the identifier 25 limits combinations of theidentified gain coefficients a1(k) hat, a2(k) hat determined in STEP5-7shown in FIG. 10 according to the first limiting condition in STEP5-8-1through STEP5-8-8.

Specifically, the identifier 25 decides whether or not the value of theidentified gain coefficient a2(k) hat determined in STEP5-7 is equal toor greater than a lower limit value A2L (see FIG. 12) for the gaincoefficient a2 in the identifying coefficient limiting range inSTEP5-8-1.

If the value of the identified gain coefficient a2(k) is smaller thanA2L, then since a point on the coordinate plane shown in FIG. 12, whichis expressed by (a1(k) hat, a2(k) hat), determined by the combination ofthe values of the identified gain coefficients a1(k) hat, a2(k) hat doesnot reside in the identifying coefficient limiting range, the value ofa2(k) hat is forcibly changed to the lower limit value A2L in STEP5-8-2.Thus, the point (a1(k) hat, a2(k) hat) on the coordinate plane shown inFIG. 12 is limited to a point in a region on and above a straight line,i.e., the straight line including the line segment Q₇Q₈, expressed by atleast a2=A2L.

Then, the identifier 25 decides whether or not the value of theidentified gain coefficient a1(k) hat determined in STEP5-7 is equal toor greater than a lower limit value A1L (see FIG. 12) for the gaincoefficient a1 in the identifying coefficient limiting range inSTEP5-8-3, and then decides whether or not the value of the identifiedgain coefficient a1(k) hat is equal to or smaller than an upper limitvalue A1H (see FIG. 12) for the gain coefficient a1 in the identifyingcoefficient limiting range in STEP5-8-5. The upper limit value A1H forthe gain coefficient a1 in the identifying coefficient limiting range isrepresented by A1H=1−A2L because it is an a1 coordinate of the point Q₈where the polygonal line |a1|+a2=1 (a1>0) and the straight line a2=A2Lintersect with each other, as shown in FIG. 12.

If the value of the identified gain coefficient a1(k) hat is smallerthan the lower limit value A1L or greater than the upper limit valueA1H, then since the point (a1(k) hat, a2(k) hat) on the coordinate planeshown in FIG. 12 does not reside in the identifying coefficient limitingrange, the value of a1(k) hat is forcibly changed to the lower limitvalue A1L or the upper limit value A1H in STEP5-8-4, STEP5-8-6.

Thus, the point (a1(k) hat, a2(k) hat) on the coordinate plane shown inFIG. 12 is limited to a region on and between a straight line, i.e., thestraight line including the line segment Q₆Q₇, expressed by a1=A1L, anda straight line, i.e., the straight line passing through the point Q₈and perpendicular to the a1 axis, expressed by a1=A1H.

The processing in STEP5-8-3 and STEP5-8-4 and the processing inSTEP5-8-5 and STEP5-8-6 may be switched around. The processing inSTEP5-8-1 and STEP5-8-2 may be carried out after the processing inSTEP5-8-3 through STEP5-8-6.

Then, the identifier 25 decides whether the present values of a1(k) hat,a2(k) hat after STEP5-8-1 through STEP5-8-6 satisfy an inequality|a1|+a2≦1 or not, i.e., whether the point (a1(k) hat, a2(k) hat) ispositioned on or below or on or above the polygonal line (including linesegments Q₅Q₆ and Q₅Q₈) expressed by the functional expression |a1|+a2=1in STEP5-8-7.

If |a1|+a2≦1, then the point (a1(k) hat, a2(k) hat) determined by thevalues of a1(k) hat, a2(k) hat after STEP5-8-1 through STEP5-8-6 existsin the identifying coefficient limiting range (including itsboundaries).

If |a1|+a2>1, then since the point (a1(k) hat, a2(k) hat) deviatesupwardly from the identifying coefficient limiting range, the value ofthe a2(k) hat is forcibly changed to a value (1−|a1(k) hat|) dependingon the value of a1(k) hat in STEP5-8-8. Stated otherwise, while thevalue of a1(k) hat is being kept unchanged, the point (a1(k) hat, a2(k)hat) is moved onto a polygonal line expressed by the functionalexpression |a1|+a2=1, i.e., onto the line segment Q₅Q₆ or the linesegment Q₅Q₈ which is a boundary of the identifying coefficient limitingrange.

Through the above processing in STEP5-8-1 through 5-8-8, the values ofthe identified gain coefficients a1(k) hat, a2(k) hat are limited suchthat the point (a1(k) hat, a2(k) hat) determined thereby resides in theidentifying coefficient limiting range. If the point (a1(k) hat, a2(k)hat) corresponding to the values of the identified gain coefficientsa1(k) hat, a2(k) hat that have been determined in STEP5-7 exists in theidentifying coefficient limiting range, then those values of theidentified gain coefficients a1(k) hat, a2(k) hat are maintained.

The value of the identified gain coefficient a1(k) hat relative to theprimary autoregressive term of the discrete-system model is not forciblychanged insofar as the value resides between the lower limit value A1Land the upper limit value A1H of the identifying coefficient limitingrange. If a1(k) hat<A1L or a1(k) hat>A1H, then since the value of theidentified gain coefficient a1(k) hat is forcibly changed to the lowerlimit value A1L which is a minimum value that the gain coefficient a1can take in the identifying coefficient limiting range or the upperlimit value A1H which is a maximum value that the gain coefficient a1can take in the identifying coefficient limiting range, the change inthe value of the identified gain coefficient a1(k) hat is minimum.Stated otherwise, if the point (a1(k) hat, a2(k) hat) corresponding tothe values of the identified gain coefficients a1(k) hat, a2(k) hat thathave been determined in STEP5-7 deviates from the identifyingcoefficient limiting range, then the forced change in the value of theidentified gain coefficient a1(k) hat is held to a minimum.

After having limited the values of the identified gain coefficientsa1(k) hat, a2(k) hat, the identifier 25 limits the identified gaincoefficient b1(k) hat according to the second limiting condition inSTEP5-8-9 through STEP5-8-12.

Specifically, the identifier 25 decides whether or not the value of theidentified gain coefficient b1(k) hat determined in STEP5-7 is equal toor greater than the lower limit value B1L in STEP5-8-9. If the lowerlimit value B1L is greater than the value of the identified gaincoefficient b1(k) hat, the value of b1(k) hat is forcibly changed to thelower limit value B1L in STEP5-8-10.

The identifier 25 decides whether or not the value of the identifiedgain coefficient b1(k) hat is equal to or smaller than the upper limitvalue B1H in STEP5-8-11. If the upper limit value B1H is smaller thanthe value of the identified gain coefficient b1(k) hat, the value ofb1(k) hat is forcibly changed to the upper limit value B1H inSTEP5-8-12.

If B1L≧b1(k) hat≦B1H, then the value of the identified gain coefficientb1(k) is maintained as it is.

Through the above processing in STEP5-8-9 through 5-8-12, the value ofthe identified gain coefficient b1(k) hat is limited to a range betweenthe lower limit value B1L and the upper limit value B1H.

After the identifier 25 has limited the combination of the values of theidentified gain coefficients a1(k) hat, a2(k) hat and the identifiedgain coefficient b1(k) hat, control returns to the sequence shown inFIG. 10.

The preceding values a1(k−1) hat, a2(k−1) hat, b1(k−1) hat of theidentified gain coefficients used for determining the identified gaincoefficients a1(k) hat, a2(k) hat, b1(k) hat in STEP5-7 shown in FIG. 10are the values of the identified gain coefficients limited according tothe first and second limiting conditions in STEP5-8 in the precedingcontrol cycle.

After having limited the identified gain coefficients a1(k) hat, a2(k)hat, b1(k) hat as described above, the identifier 25 updates the matrixP(k) according to the equation (10) for the processing of a next controlcycle in STEP5-9, after which control returns to the main routine shownin FIG. 8.

The processing subroutine of STEP5 for the identifier 25 has beendescribed above.

In FIG. 8, after the processing of the identifier 25 has been carriedout, the target air-fuel ratio generator 28 determines the gaincoefficients a1, a2, b1 in STEP6.

In STEP6, if the value of the flag f/id/cal set in STEP2 is “1”, i.e.,if the gain coefficients a1, a2, b1 have been identified by theidentifier 25, then the values of the gain coefficients a1, a2, b1 areset to the identified gain coefficients a1 hat, a2 hat, b1 hat (limitedin STEP5-8) determined by the identifier 25 in STEP5. If the value ofthe flag f/id/cal is “0”, i.e., if the gain coefficients a1, a2, b1 havenot been identified by the identifier 25, then the values of the gaincoefficients a1, a2, b1 are set to predetermined values.

Then, the target air-fuel ratio generator 28 effects a processingoperation of the estimator 26, i.e., calculates the estimateddifferential output VO2 bar, in STEP7 of the main routine shown in FIG.8.

The calculating subroutine of STEP7 is shown in detail in FIG. 14. Asshown in FIG. 14, the estimator 26 calculates the coefficients α1, α2,βj (j=1−d) to be used in the equation (13), using the gain coefficientsa1, a2, b1 determined in STEP6 (these values are basically theidentified gain coefficients a1 hat, a2 hat, b1 hat which have beenlimited in STEP5-8 shown in FIG. 10) according to the definitionaccompanying the equation (12) in STEP7-1.

Then, in STEP7-2, the estimator 26 calculates the estimated differentialoutput VO2(k+d) bar (estimated value of the differential output VO2after the total dead time d from the time of the present control cycle)according to the equation (13), using the time-series data VO2(k),VO2(k−1), from before the present control cycle, of the differentialoutput VO2 of the O₂ sensor calculated in each control cycle in STEP3shown in FIG. 8, the time-series data kact(k−j) (j=0−d1), from beforethe present control cycle, of the differential output kact of the LAFsensor 5, the time-series data kcmd(k−j) (normally, kcmd(k−j)=usl(k−j)(j=1−d2−1), from before the preceding control cycle, of the commanddifferential air-fuel ratio kcmd (=the demand differential air-fuelratio usl as limited) determined in each control cycle by the limiter 30as described later on, and the coefficients α1, α2, βj calculated asdescribed above.

Referring back to FIG. 8, the target air-fuel ratio generator 28 thencalculates the demand differential air-fuel ratio usl with the slidingmode controller 27 in STEP8.

The calculating subroutine of STEP8 is shown in detail in FIG. 15.

As shown in FIG. 15, the sliding mode controller 27 calculates a valueσ(k+d) bar (corresponding to an estimated value, after the dead time d,of the switching function σ defined according to the equation (15)),after the total dead time d from the present control cycle, of theswitching function σ bar defined according to the equation (25), usingthe time-series data VO2(k+d) bar, VO2(k+d−1) bar (present and precedingvalues of the estimated differential output VO2 bar) of the estimateddifferential output VO2 bar determined by the estimator 26 in STEP7 inSTEP8-1.

If the switching function σ bar is excessively large, then the value ofthe reaching control law input urch determined depending on the value ofthe switching function σ bar becomes excessively large, causing theadaptive control law input uadp to change abruptly. Therefore, thedemand differential air-fuel ratio usl determined by the sliding modecontroller 27 and the target air-fuel ratio KCMD tends to becomeunstable. According to the present embodiment, the value of theswitching function σ bar is set to fall in a predetermined range, and ifthe value of the σ bar determined according to the equation (25) exceedsan upper or lower limit of the predetermined range, then the value ofthe σ bar is forcibly set to the upper or lower limit of thepredetermined range.

Then, the sliding mode controller 27 accumulates values of the switchingfunction σ bar calculated in respective control cycles in STEP8-1 (moreaccurately, values produced when the value of the σ bar is multiplied bythe period (constant period) of the control cycles of the exhaust-sidecontrol unit 7 a), i.e., adds a value of the σ bar calculated in thepresent control cycle to the sum determined in the preceding controlcycle, thereby calculating an integrated value of the σ bar (whichcorresponds to the term at the right end of the equation (27)) inSTEP8-2.

In order to prevent the adaptive control law input uadp determineddepending on the integrated value of the σ bar, the integrated value ofthe σ bar is set to fall in a predetermined range, as with STEP8-1.Specifically, if the integrated value of the σ bar exceeds an upper orlower limit of the predetermined range, then the integrated value of theσ bar is limited to the upper or lower limit.

Then, the sliding mode controller 27 calculates the equivalent controlinput ueq, the reaching control law input urch, and the adaptive controllaw uadp according to the respective equations (24), (26), (27) inSTEP8-3, using the time-series data VO2(k+d) bar, VO2(k+d−1) bar of theestimated differential output VO2 bar determined by the estimator 26 inSTEP7, the value σ(k+d) bar of the switching function and its integratedvalue which are determined respectively in STEP8-1 and STEP8-2, the gaincoefficients a1, a1, b1 determined in STEP 6 (which are basically thegain coefficients a1 hat, a2 hat, b1 hat limited in STEP5-8 shown inFIG. 10).

The sliding mode controller 27 then adds the equivalent control inputueq, the reaching control law input urch, and the adaptive control lawuadp determined in STEP8-3 to calculate an input to be applied to theobject exhaust system E for converging the output signal VO2/OUT of theO₂ sensor 6 to the target value VO2/TARGET in STEP8-4.

The processing operation of the sliding mode controller 27 in STEP8 hasbeen described above.

In FIG. 8, the target air-fuel ratio generator 28 carries out a processwith the limiter 30. Prior to limiting the demand differential air-fuelratio usl calculated by the sliding mode controller 27, the limiter 30determines the stability of the status of the output signal VO2/OUT ofthe O₂ sensor 6 (the output status of the object exhaust system,hereinafter referred to as an “SLD control status”) which is controlledaccording to the adaptive sliding mode control process carried out bythe sliding mode controller 27 in STEP9.

A process of determining the stability will briefly be described priorto specifically describing the details of the process of determining thestability.

In the present embodiment, in each control cycle of the exhaust-sidecontrol unit 7 a, the sliding mode controller 27 uses a difference Δσbar (corresponding to a rate of change of the value of the switchingfunction σ bar) between the present value σ(k+d) bar of the switchingfunction σ bar calculated in STEP8-1 and a preceding value σ(k+d−1) barthereof in a preceding control cycle, and the product σ(k+d) bar·Δσ barof the switching function σ bar and the present value σ(k+d) bar, as abasic parameter for determining the stability of the SLD control status(the product σ(k+d) bar·Δσ bar will hereinafter be referred to as a“stability determining basic parameter Pstb”).

The stability determining basic parameter Pstb (=σ(k+d) bar·Δσ bar)corresponds to the time-differentiated function of a Lyapunov function σbar²/2 relative to the switching function σ bar. The state in whichPstb≦0 is basically a state in which the value of the switching functionσ bar is converged to or converging to “0” (the state quantitycomprising the time-series data VO2(k+d) bar, VO2(k+d−1) bar of theestimated differential output VO2 bar is converged to or converging tothe hyperplane σ bar=0). The state in which Pstb>0 is basically a statein which the value of the switching function σ bar is getting away from“0” (the state quantity comprising the time-series data VO2(k+d) bar,VO2(k+d−1) bar of the estimated differential output VO2 bar is gettingaway from the hyperplane σ bar=0).

Therefore, it is possible to determined whether the SLD control statusis stable or not based on whether or not the value of the stabilitydetermining basic parameter Pstb is equal to or smaller than “0”.

However, if the stability of the SLD control status is determined solelyby comparing the value of the stability determining basic parameter Pstbwith “0”, then any slight noise contained in the value of the σ bar willaffect the determined result of the stability. Furthermore, if thedetermined result of the stability is established based on the value ofthe stability determining basic parameter Pstb in each control cycle,then the determined result tends to change frequently.

In this embodiment, in each control cycle of the exhaust-side controlunit 7 a, it is temporarily determined whether the SLD control status isstable or not based on whether or not the value of the stabilitydetermining basic parameter Pstb is equal or smaller than apredetermined value ε that is of a positive value slightly greater than“0”. In addition, in a predetermined period longer than the controlcycles of the exhaust-side control unit 7 a, the frequency cnt/judst atwhich the SLD control status is temporarily determined as unstable basedon the value of the stability determining basic parameter Pstb (morespecifically, the number of control cycles in which Pstb>ε in thepredetermined period, hereinafter referred to as a “temporarily unstabledecision frequency cnt/judst”) is measured. Whether the SLD controlstatus is stable or not is determined by comparing the temporarilyunstable decision frequency cnt/judst with predetermined thresholds.

In the present embodiment, the predetermined thresholds with which thetemporarily unstable decision frequency cnt/judst is compared include afirst threshold SSTB1 and a second threshold SSTB2 (SSTB1<SSTB2). If thetemporarily unstable decision frequency cnt/judst is equal to or smallerthan the first threshold SSTB1 (cnt/judst<SSTB1), then the SLD controlstatus is determined as stable. If cnt/judst>SSTB1, then the SLD controlstatus is determined as unstable. If the SLD control status isdetermined as unstable (cnt/judst>SSTB1), then when the temporarilyunstable decision frequency cnt/judst is equal to or smaller than thesecond threshold SSTB2 (cnt/judst≦SSTB2), the level of the instabilityof the SLD control status is determined as low (such a state willhereinafter be referred to as a “low-level unstable state”), and whencnt/judst>SSTB2, the level of the instability of the SLD control statusis determined as high (such a state will hereinafter be referred to as a“high-level unstable state”). Thus, if the SLD control status isdetermined as unstable, then the level of the instability of the SLDcontrol status is also determined depending on the value of thetemporarily unstable decision frequency cnt/judst.

Based on the above principles of the determination of the stability ofthe SLD control status, the process of determining the stability of theSLD control status will be described in greater detail below.

The process of determining the stability of the SLD control status iscarried out according to a flowchart shown in FIG. 16.

The limiter 30 calculates the stability determining basic parameter Pstb(=σ(k+d) bar·Δσ bar) defined above from the present value σ(k+d) and thepreceding value σ(k+d−1) of the switching function σ bar determined bythe sliding mode controller 27 in STEP8-1 in STEP9-1 in FIG. 16.

Then, the limiter 30 determines whether the value of the timer countertm/stb (count-down timer) initialized in STEPd-11 in FIG. 7 has become“0” or not, i.e., whether the time that has elapsed from the start ofthe manipulation of the air-fuel ratio of the internal combustion engine1 based on the target air-fuel ratio KCMD generated by the targetair-fuel ratio generator 28 (the control process for converging theoutput signal VO2/OUT from the O₂ sensor 6 to the target valueVO2/TARGET, which may hereinafter be referred to as an “air-fuel ratiomanipulation exhaust system output control process”) has reached apredetermined time expressed by the initial value TMSTB of the timercounter tm/stb, in STEP9-2.

If tm/stb≠0, and hence the time that has elapsed from the start of theair-fuel ratio manipulation exhaust system output control process hasnot reached the predetermined time (:TMSTB) (immediately after start ofthe air-fuel ratio manipulation exhaust system output control process),then the stability of the SLD control status based on the measurement ofthe temporarily unstable decision frequency cnt/judst, etc. is notdetermined, and STEP9-3 is carried out, after which control returns tothe main routine shown in FIG. 8.

In STEP9-3, the value of the timer counter tm/stb (count-down timer) formeasuring the predetermined period in which to measure the temporarilyunstable decision frequency cnt/judst is set to a predetermined initialvalue TMJUDST corresponding to the time of the predetermined period. InSTEP9-3, the value of the temporarily unstable decision frequencycnt/judst is initialized to “0”. In STEP9-3, moreover, the value of aflag f/stb1 which indicates whether the SLD control status is stable ornot with “0” and “1”, respectively, is initialized to “0”, and the valueof a flag f/stb2 which indicates whether the SLD control status is inthe low-level unstable state or the high-level unstable state with “0”and “1”, respectively, is initialized to “0”.

If the value of the timer tm/stb is “0”, and hence the time that haselapsed from the start of the air-fuel ratio manipulation exhaust systemoutput control process has reached the predetermined time expressed bythe initial value TMSTB of the timer tm/stb, then the limiter 30compares the stability determining basic parameter Pstb with thepredetermined value ε (>0) in STEP9-4. If Pstb≧ε, then the SLD controlstatus is regarded temporarily stable, and the value of the temporarilyunstable decision frequency cnt/judst is maintained at its present value(preceding value) in STEP9-5. If Pstb>ε, then the SLD control status isregarded as temporarily stable, and the value of the temporarilyunstable decision frequency cnt/judst is incremented from the presentvalue (preceding value) by “1” in STEP9-6.

Then, the limiter 30 compares the value of the temporarily unstabledecision frequency cnt/judst determined in the present cycle in STEP9-5or STEP9-6 with the first threshold SSTB1 in STEP9-7. Ifcnt/judst≦SSTB1, then the SLD control status is regarded as stable, andthe flag f/stb1 (hereinafter referred to as a “stability decision flagf/stb1”) is set to “0” in STEP9-8.

If cnt/judst>SSTB1 (the SLD control status is unstable), then the limit30 compares the value of the temporarily unstable decision frequencycnt/judst with the second threshold SSTB2 in STEP9-9. Ifcnt/judst≦SSTB2, then the SLD control status is regarded as in thelow-level unstable state, and the stability decision flag f/stb1 is setto “1” in STEP9-10. If cnt/judst>SSTB2, then the SLD control status isregarded as in the high-level unstable state, and the stability decisionflag f/stb1 is set to “1” and the flag f/stb 2 (hereinafter referred toas an “instability level decision flag f/stb2”) is set to “1” inSTEP9-11.

Then, the limiter 30 decrements the value of the timer counter tm/judstfrom the present value (preceding value) by “1” in STEP9-12, andthereafter determines whether the updated value of the timer countertm/judst has reached “0” or not, i.e., whether the predetermined periodrepresented by the initial value TMJUDST of the timer counter tm/judsthas elapsed or not, in STEP9-13.

If tm/judst≠0, and hence the predetermined period (:TMJUDST) has notelapsed (the timer counter tm/judst has not run out), then controlreturns to the main routine shown in FIG. 8.

If tm/judst=0, and hence the predetermined period (:TMJUDST) has elapsed(the timer counter tm/judst has run out), then the limiter 30 determinesthe value of the stability decision flag f/stb1 in STEP9-14. Iff/stb1=1, then the value of the timer counter tm/judst is set to itsinitial value TMJUDST, and the value of the temporarily unstabledecision frequency cnt/judst is reset to “0” in STEP9-16, after whichcontrol returns to the main routine shown in FIG. 8.

If f/stb1=0 in STEP9-14, i.e., if the SLD control status in the presentcontrol cycle is determined as stable, then the value of the instabilitylevel decision flag f/stb2 is reset to “0” in STEP9-15, and STEP9-16 isexecuted, after which control returns to the main routine shown in FIG.8.

The manner in which the values of the stability determining basicparameter Pstb, the temporarily unstable decision frequency cnt/judst,the stability decision flag f/stb1, and the instability level decisionflag f/stb2 vary is illustrated respectively in first, second, third,and fourth stages of FIG. 17. In the first stage of FIG. 17, the valueof the stability determining basic parameter Pstb is shown as beingequal to or greater than “0”. However, the stability determining basicparameter Pstb may actually be of a negative value.

As shown in FIG. 17, the temporarily unstable decision frequencycnt/judst, i.e., the frequency (the number of times) at which the valueof the stability determining basic parameter Pstb is Pstb>ε, is measuredin every predetermined period (:TMJUDST) and reset to “0” each time thepredetermined period (:TMJUDST) elapses (see STEP9-16). As indicated ina period T1 or T2 in FIG. 17, if the value of the temporarily unstabledecision frequency cnt/judst does not reach the first threshold SSTB1,i.e., if a situation in which the value of the stability determiningbasic parameter Pstb exceeds the predetermined value ε does not occur oroccurs only temporarily, then the SLD control status is regarded asstable, and the value of the stability decision flag f/stb1 is set to“0” (see STEP9-7, STEP9-8).

As indicated in a period T3 in FIG. 17, if the situation in which thevalue of the stability determining basic parameter Pstb exceeds thepredetermined value ε in the predetermined period continues for acertain time or occurs frequently, and the value of the temporarilyunstable decision frequency cnt/judst exceeds the first threshold SSTB1,then the SLD control status is determined as in the low-level unstablestate, and the value of the stability decision flag f/stb1 is set to “1”(see STEP9-9, STEP9-10). In this case, as the temporarily unstabledecision frequency cnt/judst is reset in every predetermined period, thevalue of the stability decision flag f/stb1 is also reset to “0” inevery predetermined period.

As indicated in a period T4 in FIG. 17, if the situation in which thevalue of the stability determining basic parameter Pstb exceeds thepredetermined value ε in the predetermined period continues for arelatively long time or occurs frequently, and the value of thetemporarily unstable decision frequency cnt/judst exceeds the secondthreshold SSTB2, then the SLD control status is determined as in thehigh-level unstable state, and the value of the stability decision flagf/stb1 is set to “1” and the value of the instability level decisionflag f/stb2 is set to “1” (see STEP9-11). In this case, the value of theinstability level decision flag f/stb2 is kept at “1” regardless of thevalue of the temporarily unstable decision frequency cnt/judst during anext predetermined period (period T5). Only if the value of thetemporarily unstable decision frequency cnt/judst is continuously keptequal to or smaller than the first threshold SSTB1 and the value of thestability decision flag f/stb1 is kept at “0” during the nextpredetermined period (period T5), the instability level decision flagf/stb2 is reset to “0” at the end of the period T5(STEP9-14, STEP9-15).Therefore, if the SLD control status is determined as in the high-levelunstable state within a certain predetermined period, the value of theinstability level decision flag f/stb2 is kept at “1” indicative of thehigh-level unstable state within a next predetermined period unless thevalue of the temporarily unstable decision frequency cnt/judst iscontinuously kept equal to or smaller than the first threshold SSTB1, sothat the decision of the high-level unstable state essentiallycontinues.

Referring back to FIG. 8, after the limiter 30 in the target air-fuelratio generator 28 determines the stability of the SLD control status asdescribed above, the limiter 30 limits the demand differential air-fuelratio usl calculated by the sliding mode controller 27 in STEP8 todetermine the command differential air-fuel ratio kcmd in STEP10.

Prior to specifically describing the limiting process in detail,allowable ranges used by the limiting process will be described below.

The limiting process carried out by the limiter 30 forcibly sets thecommand differential air-fuel ratio kcmd to an upper or lower limit ofan allowable range if the demand differential air-fuel ratio usldeviates from the allowable range beyond the upper or lower limitthereof, in order to keep the command differential air-fuel ratio kcmdwhich defines the target air-fuel ratio KCMD to be finally given to theengine-side control unit 7 b. If the demand differential air-fuel ratiousl falls in the allowable range, the limiting process sets up thedemand differential air-fuel ratio usl directly as the commanddifferential air-fuel ratio kcmd.

In the present embodiment, the limiting process uses a plurality ofallowable ranges as shown in FIG. 18. Specifically, the allowable rangesinclude an allowable range for unstable low level used to limit thedemand differential air-fuel ratio usl when the determined stability ofthe SLD control status represents the low-level unstable state (f/stb1=1and f/stb2=0) (except when the internal combustion engine 1 is idling)in STEP9, and an allowable range for unstable high level used to limitthe demand differential air-fuel ratio usl when the determined stabilityof the SLD control status represents the high-level unstable state(f/stb2=1).

The allowable ranges for the limiting process also include an allowablerange after FC (fuel cut-off) used to limit the demand differentialair-fuel ratio usl immediately after the fuel supply to the internalcombustion engine 1 is cut off (specifically, until a predetermined timeelapses after the fuel supply to the internal combustion engine 1 is cutoff) except when the determined stability of the SLD control statusrepresents the low-level unstable state or the high-level unstablestate, an allowable range after start used to limit the demanddifferential air-fuel ratio usl immediately after the internalcombustion engine 1 starts (more specifically, until a predeterminedtime elapses after the internal combustion engine 1 starts), and anallowable range after operation with lean mixture used to limit thedemand differential air-fuel ratio usl immediately after the operationmode of the internal combustion engine 1 changes from the lean operationmode in which the air-fuel ratio of the internal combustion engine 1 isa lean ratio to the normal operation mode in which the air-fuel ratiomanipulation exhaust system output control process is performed(specifically, until a predetermined time elapses after the leanoperation mode is ended). In this embodiment, the allowable range afterFC, the allowable range after start, and the allowable range afteroperation with lean mixture are identical allowable ranges, and willcollectively be referred to as an allowable range afterFC/start/operation with lean mixture.

The allowable ranges for limiting the demand differential air-fuel ratiousl further include an allowable range after being driven under loadused to limit the demand differential air-fuel ratio usl immediatelyafter a vehicle incorporating the internal combustion engine 1 starts(specifically, until a predetermined time elapses after the internalcombustion engine 1 starts to operate drive wheels of the vehicle whichserves as a load on the internal combustion engine 1), except when thedetermined stability of the SLD control status represents the low-levelunstable state or the high-level unstable state, and except immediatelyafter the fuel supply to the internal combustion engine 1 is cut off, orthe internal combustion engine 1 starts, or the internal combustionengine 1 operates with lean mixture, and an allowable range idling usedto limit the demand differential air-fuel ratio usl while the internalcombustion engine 1 is idling, except when the determined stability ofthe SLD control status represents the high-level unstable state, andexcept immediately after the fuel supply to the internal combustionengine 1 is cut off, or the internal combustion engine 1 starts, or theinternal combustion engine 1 operates with lean mixture.

The allowable ranges for limiting the demand differential air-fuel ratiousl also include an adaptive allowable range used to limit the demanddifferential air-fuel ratio usl while the internal combustion engine 1is operates normally except those states corresponding to the abovevarious allowable ranges.

The allowable range for unstable low level is a fixed allowable rangewith its upper and lower limits set respectively to predetermined fixedvalues H, L (the predetermined value H will hereinafter be referred toas an “upper-limit third predetermined value H” and the predeterminedvalue L will hereinafter be referred to as a “lower-limit fourthpredetermined value L”). The allowable range for unstable low level hasa standard (intermediate) extent at the upper and lower limits thereof,among the various allowable ranges.

The allowable range for unstable high level is a fixed allowable rangewith its upper and lower limits set respectively to predetermined fixedvalues STABH, STABL (the predetermined value STABH will hereinafter bereferred to as an “upper-limit first predetermined value STABH” and thepredetermined value STABL will hereinafter be referred to as a“lower-limit first predetermined value STABL”). The allowable range forunstable high level has a narrowest extent at the upper and lower limitsthereof, among the various allowable ranges.

The allowable range for idling is a fixed allowable range with its upperand lower limits set respectively to predetermined fixed values HI, LI(the predetermined value HI will hereinafter be referred to as an“upper-limit second predetermined value HI” and the predetermined valueLI will hereinafter be referred to as a “lower-limit third predeterminedvalue LI”). The allowable range for idling has a relatively narrowextent at the upper and lower limits thereof (narrower than theallowable range for unstable low level).

The allowable range after FC/start/operation with lean mixture is anallowable range with its upper limit set to a predetermined fixed valueAFCH (hereinafter referred to as an “upper-limit fifth predeterminedvalue AFCH”) and its lower limit sequentially variable (in each controlcycle) depending on how the demand differential air-fuel ratio usldeviates from the allowable range, between the lower-limit firstpredetermined value STABL and a predetermined value LL smaller than thelower-limit first predetermined value STABL (hereinafter referred to asa “lower-limit fifth predetermined value LL”). The lower limit of theallowable range after FC/start/operation with lean mixture is widestamong the various allowable ranges. While the internal combustion engine1 is idling, the lower limit of the allowable range afterFC/start/operation with lean mixture is not variable, but set to thelower-limit third predetermined value LI which is the lower limit of theallowable range for idling.

The allowable range after being driven under load is an allowable rangewith its lower limit set to a predetermined fixed value VSTL(hereinafter referred to as a “lower-limit second predetermined valueVSTL”) and its upper limit sequentially variable (in each control cycle)depending on how the demand differential air-fuel ratio usl deviatesfrom the allowable range, between the upper-limit first predeterminedvalue STABH and a predetermined value HH larger than the upper-limitfirst predetermined value STABH (hereinafter referred to as an“upper-limit fourth predetermined value HH”). The lower limit of theallowable range after being driven under load is relatively narrow(narrower than the lower limit of the allowable range for idling).

The adaptive allowable range is an allowable range with its upper limitsequentially variable (in each control cycle) depending on how thedemand differential air-fuel ratio usl deviates from the allowablerange, between the upper-limit first predetermined value STABH and theupper-limit fourth predetermined value HH and its lower limitsequentially variable (in each control cycle) depending on how thedemand differential air-fuel ratio usl deviates from the allowablerange, between the lower-limit first predetermined value STABL and thelower-limit fifth predetermined value LL.

Since the demand differential air-fuel ratio usl to be limited to thevarious allowable ranges represents the difference between the air-fuelratio of the internal combustion engine 1 and the air-fuel ratioreference value FLAF/BASE, the upper and lower limits of the allowableranges represent differences with the air-fuel ratio reference valueFLAF/BASE, i.e., positive and negative values with respect to theair-fuel ratio reference value FLAF/BASE represented as “0”. Theupper-limit first through fifth predetermined values STABH, HI, H, HH,AFCH are related in magnitude with each other as 0<STABH<HI<H<HH<AFCH asshown in FIG. 18. Similarly, the lower-limit first through fifthpredetermined values STABL, VSTL, LI, L, LL are related in magnitudewith each other as 0>STABL>VSTL>LI>L>LL as shown in FIG. 18.

In view of the above definition of the various allowable ranges, thelimiting process in STEP10 will be described in specific detail below.The limiting process is carried out according to a flowchart shown inFIG. 19. An allowable range for the limiting process which isestablished in an idling state of the internal combustion engine 1,which allowable range is not limited to the above allowable range foridling, has an upper limit AHFI and a lower limit ALFI which will bereferred to respectively as an idling upper limit AHFI and an idlinglower limit ALFI. An allowable range for the limiting process which isestablished in an operating state of the internal combustion engine 1except for the idling state has an upper limit AHF and a lower limit ALFwhich will be referred to respectively as a non-idling upper limit AHFand a non-idling lower limit ALF. The variable upper and lower limits ofthe adaptive allowable range will be referred to respectively as anadaptive upper limit ah and an adaptive lower limit al.

As shown in FIG. 19, the limiter 30 carries out a process of determiningan allowable range for limiting the demand differential air-fuel ratiousl in the present cycle in STEP10-1.

The process of determining an allowable range in STEP10-1 is performedaccording to a flowchart shown in FIG. 20.

First, the limiter 30 determines the value of the instability leveldecision flag f/stb2 established in the process of determining thestability of the SLD control status in STEP10-1-1. If f/stb2=1, i.e., ifthe SLD control status is in the high-level unstable state, then thelimiter 30 sets the idling upper limit AHFI, the non-idling upper limitAHF, and the adaptive upper limit ah (more specifically, a present valueah(k−1) of the adaptive upper limit ah to the upper-limit firstpredetermined value STABH which is the upper limit of the allowablerange for unstable high level (see FIG. 18), and also sets the idlinglower limit ALFI, the non-idling lower limit ALF, and the adaptive lowerlimit al (more specifically, a present value al(k−1) of the adaptivelower limit al) to the lower-limit first predetermined value STABL whichis the lower limit of the allowable range for unstable high level inSTEP10-1-2. After STEP10-1-2, control returns to the routine shown inFIG. 19.

In STEP10-1-2, the allowable range for limiting the demand differentialair-fuel ratio usl is set to the allowable range for unstable highlevel, i.e., the narrowest fixed allowable range, regardless of theoperating state of the internal combustion engine 1. The adaptiveallowable range which is made variable as described later on isinitialized to the allowable range for unstable high level.

If f/stb2=0 in STEP10-1-1, then the limiter 30 determines the stabilitydecision flag f/stb1 established in the process of determining thestability of the SLD control status in STEP10-1-3. If f/stb1=1, i.e., ifthe SLD control status is in the low-level unstable state, then thelimiter 30 sets the idling upper limit AHFI to the upper-limit secondpredetermined value HI which is the upper limit of the idling allowablerange (see FIG. 18) and also sets the idling lower limit ALFI to thelower-limit third predetermined value LI which is the lower limit of theidling allowable range in STEP10-1-4. In STEP10-1-4, the limiter 30 alsosets the non-idling upper limit AHF and the present value ah(k−1) of theadaptive upper limit ah to the upper-limit third predetermined value Hwhich is the fixed upper limit of the allowable range for unstable lowlevel (see FIG. 18), and also sets the non-idling lower limit ALF andthe adaptive lower limit al (al(k−1)) to the lower-limit fourthpredetermined value L which is the fixed lower limit of the allowablerange for unstable low level. After STEP10-1-4, control returns to theroutine shown in FIG. 19.

In STEP10-1-4, the allowable range for limiting the demand differentialair-fuel ratio usl is set to the allowable range for unstable low level(the fixed standard range) in an operating state of the internalcombustion engine 1 except for the idling state, and set to theallowable range for idling (fixed narrow range) which is narrower thanthe allowable range for unstable low level in the idling state of theinternal combustion engine 1. The adaptive allowable range isinitialized to the allowable range for unstable low level.

If f/stb1=0 in STEP10-1-3, i.e., if the SLD control status is stable inSTEP9, then the limiter 30 determines whether the internal combustionengine 1 is in a state immediately after the supply of fuel is cut offor not, i.e., whether the time that has elapsed after the end of thecutting-off of the supply of fuel has not reached a predetermined timeor not, in STEP10-1-5.

The decision in STEP10-1-5 is carried out as follows: In thisembodiment, data indicative of whether the supply of fuel to theinternal combustion engine 1 is cut off or not is given from theengine-side control unit 7 b to the exhaust-side control unit 7 a. Thelimiter 30 activates a timer (not shown) from the time when itrecognizes the end of the cutting-off of the supply of fuel based on thedata, for thereby measuring the time that has elapsed from the time ofthe end of the cutting-off of the supply of fuel. The limiter 30determines a period until the elapsed time reaches a predetermined time(fixed value) as the state immediately after the supply of fuel is cutoff.

In this embodiment, while the supply of fuel is being cut off, themanipulation of the air-fuel ratio based on the target air-fuel ratioKCMD generated by the target air-fuel ratio generator 28, i.e., thecontrol process of converging the output signal VO2/OUT from the O₂sensor 6 to the target value VO2/TARGET, is not performed (see FIG. 7showing details of STEPd in FIG. 6), but the processing in the targetair-fuel ratio generator 28 for calculating the demand differentialair-fuel ratio usl and the target air-fuel ratio KCMD is carried out. Inthe present embodiment, the state in which the supply of fuel to theinternal combustion engine 1 is being cut off is determined as the stateimmediately after the supply of fuel is cut off.

If the internal combustion engine 1 is in the state immediately afterthe supply of fuel is cut off (including the state in which the supplyof fuel to the internal combustion engine 1 is being cut off) inSTEP10-1-5, then the limiter 30 sets the idling upper limit AHFI and anon-idling upper limit AHF to the upper-limit fifth predetermined valueAFCH which is the upper limit of the allowable range afterFC/start/operation with lean mixture (see FIG. 18) in STEP10-1-6. InSTEP10-1-6, the limiter 30 also sets the idling lower limit ALFI to thelower-limit third predetermined value LI which is the lower limit of theallowable range for idling and sets the non-idling lower limit ALF tothe present value of the adaptive lower limit al (the adaptive lowerlimit al(k−1) determined in the preceding control cycle). AfterSTEP10-1-6, control goes back to the routine shown in FIG. 19.

In STEP10-1-6, the allowable range for limiting the demand differentialair-fuel ratio usl is set to the allowable range afterFC/start/operation with lean mixture whose lower limit is fixed to thelower-limit third predetermined value LI which is the lower limit of theallowable range for idling if the internal combustion engine 1 is in theidling state. If the internal combustion engine 1 is in an operatingstate other than the idling state, then the allowable range for limitingthe demand differential air-fuel ratio usl is set to the allowable rangeafter FC/start/operation with lean mixture whose lower limit is set tothe sequentially variable adaptive lower limit al. At any rate, theallowable range for limiting the demand differential air-fuel ratio uslis set to a range which is widest at its upper limit, or more generally,a range whose upper limit is not lower than the upper-limit fifthpredetermined value AFCH.

If the internal combustion engine 1 is riot in the state immediatelyafter the supply of fuel is cut off in STEP10-1-5, then the limiter 30determines whether the internal combustion engine 1 is in a stateimmediately after it starts or not, i.e., whether the time that haselapsed after the start of the internal combustion engine 1 (morespecifically, after the confirmation of the full fuel combustion in theinternal combustion engine 1) does not reach a predetermined time ornot, in STEP10-1-7.

In this case, data indicative of whether the internal combustion engine1 starts or not, i.e., whether the full fuel combustion in the internalcombustion engine 1 is confirmed or not, is given from the engine-sidecontrol unit 7 b to the exhaust-side control unit 7 a. The limiter 30activates a timer (not shown) from the time when it recognizes the startof the internal combustion engine 1 based on the data, for therebymeasuring the time that has elapsed from the start of the internalcombustion engine 1. The limiter 30 determines a period until theelapsed time reaches a predetermined time (fixed value) as the stateimmediately after the internal combustion engine 1 starts.

If the internal combustion engine 1 is in the state immediately after itstarts in STEP10-1-7, then the limiter 30 carries out STEP10-1-6 to setthe idling upper limit AHFI, the non-idling upper limit AHF, the idlinglower limit ALFI, and the non-idling lower limit ALF as described above,after which control returns to the routine shown in FIG. 19.

If the internal combustion engine 1 is not in the state immediatelyafter it starts in STEP10-1-7, then the limiter 30 determines whetherthe internal combustion engine 1 is in a state immediately after the endof its lean operation mode or not, i.e., whether the time that haselapsed after the operation mode of the internal combustion engine 1changes from the lean operation mode to the normal operation mode hasreached a predetermined time or not, in STEP10-1-8.

In this case, data indicative of whether the internal combustion engine1 is in the lean operation mode or not is given from the engine-sidecontrol unit 7 b to the exhaust-side control unit 7 a. The limiter 30activates a timer (not shown) from the time when it recognizes the endof the lean operation mode based on the data, for thereby measuring thetime that has elapsed from the start of the timer. The limiter 30determines a period until the elapsed time reaches a predetermined time(fixed value) as the state immediately after the end of its leanoperation mode, more specifically, the state immediately after theoperation mode of the internal combustion engine 1 changes from the leanoperation mode to the normal operation mode.

If the internal combustion engine 1 is in the state immediately afterthe end of its lean operation mode in STEP10-1-8, then the limiter 30carries out STEP10-1-6 to set the idling upper limit AHFI, thenon-idling upper limit AHF, the idling lower limit ALFI, and thenon-idling lower limit ALF as described above, after which controlreturns to the routine shown in FIG. 19.

If the internal combustion engine 1 is not in the state immediatelyafter the end of its lean operation mode in STEP10-1-8, then the limiter30 determines whether the internal combustion engine 1 is in its idlingstate or not in STEP10-1-9.

In this case, data indicative of whether the internal combustion engine1 is in its idling state or not is given from the engine-side controlunit 7 b to the exhaust-side control unit 7 a. The limiter 30 determineswhether the internal combustion engine 1 is in its idling state or notbased on the data.

If the internal combustion engine 1 is in the idling state (at thistime, the SLD control status is not in the low-level unstable state orthe high-level unstable state, or the internal combustion engine 1 isnot in the state immediately after the supply of fuel is cut off, orimmediately after it starts, or immediately after the end of the leanoperation mode), then the limiter 30 sets the idling upper limit AHFI tothe upper-limit second predetermined value HI which is the upper limitof the allowable range for idling, and also sets the idling lower limitALFI to the lower-limit third predetermined value LI which is the lowerlimit of the allowable range for idling in STEP10-1-10. Thereafter,control returns to the routine shown in FIG. 19.

In STEP10-1-10, the allowable range for limiting the demand differentialair-fuel ratio usl is set to the fixed allowable range for idling.

If the internal combustion engine 1 is not in the idling state (at thistime, the SLD control status is not in the low-level unstable state orthe high-level unstable state, or the internal combustion engine 1 isnot in the state immediately after the supply of fuel is cut off, orimmediately after it starts, or immediately after the end of the leanoperation mode, or the internal combustion engine 1 is not idling), thenthe limiter 30 performs a process of limiting the present value ah(k−1)(the value determined in the preceding control cycle) of the adaptiveupper limit ah to a value in a range between the upper-limit firstpredetermined value STABH and the upper-limit fourth predetermined valueHH (see FIG. 18), i.e., a process of limiting the adaptive upper limitah, in STEP10-1-11 through STEP10-1-14 described below.

If the present value ah(k−1) of the adaptive upper limit ah is smallerthan the upper-limit first predetermined value STABH in STEP10-1-11,then the limiter 30 sets the present value ah(k−1) forcibly to theupper-limit first predetermined value STABH in STEP10-1-12. If thepresent value ah(k−1) of the adaptive upper limit ah is greater than theupper-limit fourth predetermined value HH in STEP10-1-13, then thelimiter 30 sets the present value ah(k−1) forcibly to the upper-limitfourth predetermined value HH in STEP10-1-14. Otherwise, the limiter 30maintains the present value ah(k−1) of the adaptive upper limit ah.

Similarly, the limiter 30 perform a process of limiting the presentvalue al(k−1) of the adaptive lower limit al to a value in a rangebetween the lower-limit first predetermined value STABL and thelower-limit fifth predetermined value LL (see FIG. 18), i.e., a processof limiting the adaptive lower limit al in STEP10-1-15 throughSTEP10-1-18. The values al , STABL, LL are negative values.

After having limited the ranges for the present upper value ah(k−1) andthe present lower value al(k−1), the limiter 30 sets the non-idlingupper limit AHF and the non-idling lower limit ALF respectively to theadaptive upper limit ah(k−1) and the adaptive lower limit al(k−1) inSTEP10-1-19.

The limiter 30 then determines whether the vehicle is in a stateimmediately after its start or not, i.e., whether the time that haselapsed after the internal combustion engine 1 starts to operate drivewheels of the vehicle which serves as a load on the internal combustionengine 1 has not reached a predetermined time or not, in STEP10-1-20.

The decision in STEP10-1-20 is carried out as follows: In the presentembodiment, the data indicative of whether the internal combustionengine 1 is in the idling state and also data indicative of the speed ofthe vehicle are given from the engine-side control unit 7 b to theexhaust-side control unit 7 a. Based on these data, the limiter 30recognizes a state in which the internal combustion engine 1 is idlingand the vehicle speed is substantially “0”, i.e., a parked or stoppedstate of the vehicle, and recognizes a start of the vehicle when thevehicle speed exceeds a predetermined value (sufficiently small value)from the parked or stopped state of the vehicle. The limiter 30 measuresthe time that has elapsed from the recognized start of the vehicle witha timer (not shown), and recognizes a period until the elapsed timereaches a predetermined time as the state immediately after the start ofthe vehicle.

If the vehicle is in the state immediately after its start inSTEP10-1-20, then the limiter 30 sets the non-idling lower limit ALF tothe lower-limit second predetermined value VSTL which is the lower limitof the allowable range after being driven under load (see FIG. 18) inSTEP10-1-21, after which control returns to the routine shown in FIG.19.

If the vehicle is not in the state immediately after its start inSTEP10-1-20, then the limiter 30 maintains the non-idling upper limitAHF and the non-idling lower limit ALF established in STEP10-1-19, afterwhich control returns to the routine shown in FIG. 19.

In STEP10-1-19 through STEP10-1-21, if the SLD control status is not inthe low-level unstable state or the high-level unstable state, or theinternal combustion engine 1 is not in the state immediately after thesupply of fuel is cut off, or immediately after it starts, orimmediately after the end of the lean operation mode, or the internalcombustion engine 1 is not idling, and if the vehicle is not in thestate immediately after its start (in a normal case), the allowablerange for limiting the demand differential air-fuel ratio usl is set tothe adaptive allowable range whose upper limit (adaptive upper limit ah)and lower limit (adaptive lower limit al) are sequentially varied, asdescribed later on. If the vehicle is in the state immediately after itsstart, then the allowable range for limiting the demand differentialair-fuel ratio usl is set to the allowable range after being drivenunder load, i.e., a range whose lower limit is relative narrow, or moregenerally, a range whose lower limit is not smaller than the lower-limitsecond predetermined value VSTL.

Referring back to FIG. 19, after the allowable range for limiting thedemand differential air-fuel ratio usl has been determined, the limiter30 determines whether the internal combustion engine 1 is in its idlingstate or not in STEP10-2.

If the internal combustion engine 1 is in its idling state, then thelimiter 30 limits the demand differential air-fuel ratio usl to theallowable range (normally the allowable range for idling) determined bythe idling upper limit AHFI and the idling lower limit ALFI which havebeen set in either one of STEP10-1-2, STEP10-1-4, STEP10-1-6, andSTEP10-1-10 (normally STEP10-1-10), for thereby determining the commanddifferential air-fuel ratio kcmd in STEP10-3 through STEP10-7.

Specifically, if the demand differential air-fuel ratio usl generated bythe sliding mode controller 27 in STEP8 (see FIG. 8) deviates from theallowable range beyond the idling lower limit ALFI (negative)(usl<ALFI), then the limiter 30 limits the value of the demanddifferential air-fuel ratio usl to the idling lower limit ALFI inSTEP10-3 and STEP10-4. If the demand differential air-fuel ratio usldeviates from the allowable range beyond the idling upper limit AHFI(positive) (usl>AHFI), then the limiter 30 limits the value of thedemand differential air-fuel ratio usl to the idling upper limit AHFI inSTEP10-5 and STEP10-6. If the demand differential air-fuel ratio uslfalls in the allowable range (ALFI≦usl≦AHFI), then the limiter 30 setsthe demand differential air-fuel ratio usl as the command differentialair-fuel ratio kcmd in STEP10-7.

If the demand differential air-fuel ratio usl deviates from theallowable range between the idling upper limit AHFI and the idling lowerlimit ALFI in STEP10-3 or STEP10-5, then in order to prevent the valueof the adaptive control law input uadp determined while the sliding modecontroller 27 is generating the demand differential air-fuel ratio uslfrom becoming larger than necessary, the limiter 30 sets the integratedvalue of the switching function σ bar (see STEP8-2 in FIG. 15) forciblyto the integrated value determined in the preceding control cycle inSTEP10-8. If the limiter 30 limits the demand differential air-fuelratio usl while the internal combustion engine 1 is idling, the presentvalues of the adaptive upper and lower limits ah, al are maintained inSTEP10-9. After STEP10-3 through STEP10-9, control returns to the mainroutine shown in FIG. 8.

If the internal combustion engine 1 is not in its idling state, then thelimiter 30 limits the demand differential air-fuel ratio usl to theallowable range (normally the adaptive allowable range) determined bythe non-idling upper limit AHF and the non-idling lower limit ALF whichhave been set in either one of STEP10-1-2, STEP10-1-4, STEP10-1-6,STEP10-1-19, and STEP10-1-21 (normally STEP10-1-19), for therebydetermining the command differential air-fuel ratio kcmd in STEP10-10through STEP10-14.

Specifically, if the demand differential air-fuel ratio usl generated bythe sliding mode controller 27 in STEP8 (see FIG. 8) deviates from theallowable range beyond the non-idling lower limit ALF (negative)(usl<ALF), then the limiter 30 limits the value of the demanddifferential air-fuel ratio usl to the non-idling lower limit ALF inSTEP10-10 and STEP10-11. If the demand differential air-fuel ratio usldeviates from the allowable range beyond the non-idling upper limit AHF(positive) (usl>AHF), then the limiter 30 limits the value of the demanddifferential air-fuel ratio usl to the non-idling upper limit AHF inSTEP10-12 and STEP10-13. If the demand differential air-fuel ratio uslfalls in the allowable range (ALF≦usl≦AHF), then the limiter 30 sets thedemand differential air-fuel ratio usl as the command differentialair-fuel ratio kcmd in STEP10-14.

In case the demand differential air-fuel ratio usl is limited by theallowable range determined by the non-idling upper limit AHF and thenon-idling lower limit ALF, the limiter 30 then updates (changes) thevalues of the adaptive upper and lower limits ah, al relative to theadaptive allowable range in STEP10-15 through STEP10-19.

Specifically, if the demand differential air-fuel ratio usl falls in theallowable range in STEP10-10 and STEP10-12, i.e., if STEP10-14 isexecuted, then the limiter 30 adds a predetermined change ΔDEC (>0,hereinafter referred to as a “decreasing unit change ΔDEC”) to thepresent value al(k−1) of the adaptive lower limit al for therebydetermining a new adaptive lower limit al(k) in STEP10-15. In STEP10-15,the limiter 30 also subtracts the decreasing unit change ΔDEC from thepresent value ah(k−1) of the adaptive upper limit ah for therebydetermining a new adaptive upper limit ah(k).

If the demand differential air-fuel ratio usl deviates from theallowable range beyond the non-idling lower limit ALF (negative), i.e.,if STEP10-11 is executed, then the limiter 30 determines whether thevehicle is in a state immediately after its start or not in STEP10-16.If the vehicle is not in the state immediately after its start, then thelimiter 30 subtracts a predetermined change ΔINC (>0, hereinafterreferred to as an “increasing unit change ΔINC”) from the present valueal(k−1) of the adaptive lower limit al for thereby determining a newadaptive lower limit a1(k) in STEP10-17. In STEP10-17, the limiter 30also subtracts the decreasing unit change ΔDEC from the present valueah(k−1) of the adaptive upper limit ah for thereby determining a newadaptive upper limit ah(k).

If the vehicle is in the state immediately after its start in STEP10-16,then the present values of the adaptive upper and lower limits ah, alare maintained in STEP10-18.

In the present embodiment, the decreasing unit change ΔDEC and theincreasing unit change ΔINC are related to each other such thatΔDEC<ΔINC.

If the demand differential air-fuel ratio usl deviates from theallowable range beyond the non-idling upper limit AHF (positive), thenthe limiter 30 determines whether the internal combustion engine 1 is inthe state immediately after the supply of fuel is cut off or not(including while the supply of fuel is being cut off or not), or whetherthe internal combustion engine 1 is in the state immediately after itstarts or not, or whether the internal combustion engine 1 is in thestate immediately after the end of its lean operation mode or not inSTEP10-19. If the internal combustion engine 1 is not in any of thesestates, then the limiter 30 adds the increasing unit change ΔINC to thepresent value ah(k−1) of the adaptive upper limit ah for therebydetermining a new adaptive upper limit ah(k) in STEP10-20. In STEP10-20,the limiter 30 also adds the decreasing unit change ΔDEC to the presentvalue al(k−1) of the adaptive lower limit al for thereby determining anew adaptive lower limit al(k).

If the internal combustion engine 1 is in either one of the states inSTEP10-19, then the present values of the adaptive upper and lowerlimits ah, al are maintained in STEP10-21.

If the demand differential air-fuel ratio usl deviates from theallowable range determined by the non-idling upper limit AHF and thenon-idling lower limit ALF in STEP10-10 or STEP10-12, then the limiter30 sets the integrated value of the switching function σ bar (seeSTEP8-2 in FIG. 15) forcibly to the integrated value determined in thepreceding control cycle in STEP10-22. After STEP10-22, control returnsto the main routine shown in FIG. 8.

In the present embodiment, the initial values of the adaptive upper andlower limits ah, al when the exhaust-side control unit 7 a is activated,i.e., when the vehicle starts operating, are respectively thelower-limit fourth predetermined value L and the upper-limit thirdpredetermined value H. That is, the initial range of the adaptiveallowable range is the same as the allowable range for unstable lowlevel.

In the present embodiment, if the SLD control status is in the low-levelor high-level unstable state when the internal combustion engine 1 is inan operating state other than the idling state, the adaptive upper andlower limits ah, al are updated in STEP10-15, STEP10-17, STEP10-20. Inthe above unstable state, the adaptive upper and lower limits ah, al areforcibly set to fixed predetermined values, i.e., the upper-limit firstpredetermined value STABH, etc. corresponding to the allowable range forunstable low level and the allowable range for unstable high level, bythe processing of STEP10-1-2 or STEP10-1-4 (see FIG. 20) in a nextcontrol cycle. Therefore, if the SLD control status is in the low-levelor high-level unstable state, then the processing of STEP10-15,STEP10-17, STEP10-20 may be omitted.

The command differential air-fuel ratio kcmd determined in each controlcycle by the limiting process in STEP10 is stored in a time-seriesfashion in a memory (not shown), and will be used for the aboveprocessing operation of the estimator 26.

According to the limiting process in STEP10, while the internalcombustion engine 1 is in the idling state, either one of the fixedallowable ranges including the allowable range for unstable low level,the allowable range for unstable high level, the allowable range afterFC/start/operation with lean mixture (specifically, the allowable rangeafter FC/start/operation with lean mixture whose lower limit is fixed tothe lower-limit third predetermined value LI (see FIG. 18)), and theallowable range for idling, is established as the allowable range forlimiting the demand differential air-fuel ratio usl (normally theallowable range for idling is established) depending on the SLD controlstatus and the operating state of the internal combustion engine 1. Thedemand differential air-fuel ratio usl is limited by the establishedallowable range to determine the command differential air-fuel ratiokcmd.

When the internal combustion engine 1 is in an operating state otherthan the idling state, either one of the allowable ranges including theallowable range for unstable low level, the allowable range for unstablehigh level, the allowable range after FC/start/operation with leanmixture (specifically, the allowable range after FC/start/operation withlean mixture whose lower limit is set to the variable adaptive lowerlimit al), the allowable range after being driven under load, and theadaptive allowable range is established as the allowable range forlimiting the demand differential air-fuel ratio usl (normally theadaptive allowable range is established) depending on the SLD controlstatus and the operating state of the internal combustion engine 1. Thedemand differential air-fuel ratio usl is limited by the establishedallowable range to determine the command differential air-fuel ratiokcmd.

Regardless of whether the internal combustion engine 1 is in anoperating state other than the idling state or in the idling state, whenthe determined stability of the SLD control status is in the high-levelunstable state, the allowable range for limiting the demand differentialair-fuel ratio usl is set to the narrowest allowable range for unstablehigh level whose upper limit AHF or AHFI and lower limit ALF or ALFI areset respectively to the fixed upper-limit first predetermined valueSTABH and the fixed lower-limit first predetermined value STABL inSTEP10-1-2.

If the SLD control status is in the low-level unstable state and theinternal combustion engine 1 is in an operating state other than theidling state, then the allowable range for limiting the demanddifferential air-fuel ratio usl is set to the allowable range forunstable low level, which has a standard extent, whose upper and lowerlimits AHF, ALF are set respectively to the upper-limit thirdpredetermined value H and the lower-limit fourth predetermined value Lin STEP10-1-4.

If the SLD control status is not in the highlevel unstable state, or theinternal combustion engine 1 is not in the state immediately after thesupply of fuel is cut off, or immediately after it starts, orimmediately after the end of the lean operation mode, or the vehicle isnot in the state immediately after it starts, and the internalcombustion engine 1 is in the idling state, then the allowable range forlimiting the demand differential air-fuel ratio usl is set to therelatively narrow allowable range for idling whose upper and lowerlimits AHFI, ALFI are set respectively to the fixed upper-limit secondpredetermined value HI and the fixed lower-limit third predeterminedvalue LI in STEP10-1-4, STEP10-1-10.

If in the normal state, i.e., if the SLD control status is not in thehigh-level unstable state or in the low-level unstable state, or theinternal combustion engine 1 is not in the state immediately after thesupply of fuel is cut off, or immediately after it starts, orimmediately after the end of the lean operation mode, or not in theidling state, or the vehicle is not in the state immediately after itstarts, then the allowable range for limiting the demand differentialair-fuel ratio usl is set to the adaptive allowable range whose upperand lower limits AHF, ALF are set respectively to the adaptive upper andlower limits ah, al in STEP10-1-19.

In the adaptive allowable range, if the demand differential air-fuelratio usl is in the adaptive allowable range (at this time, kcmd=usl),then after the limiting process, the adaptive upper and lower limits ah,al of the adaptive allowable range are updated in such a direction thatthey decrease by the decreasing unit change ΔDEC in each control cycleto respective limits provided by the upper-limit first predeterminedvalue STABH and the lower-limit first predetermined value STABL inSTEP10-15.

If the demand differential air-fuel ratio usl deviates from the adaptiveallowable range beyond the upper limit thereof (at this time,kcmd=ah(k−1), then after the limiting process, the adaptive upper limitah is updated in such a direction that the upper limit of the adaptiveallowable range increases, by the increasing unit change ΔINC in eachcontrol cycle to a limit provided by the upper-limit fourthpredetermined value HH in STEP10-20. At the same time, the adaptivelower limit al is updated in such a direction that the lower limit ofthe adaptive allowable range decreases, by the decreasing unit changeΔDEC in each control cycle to a limit provided by the lower-limit firstpredetermined value STABL.

Similarly, if the demand differential air-fuel ratio usl deviates fromthe adaptive allowable range beyond the lower limit thereof (at thistime, kcmd=al(k−1), then after the limiting process, the adaptive lowerlimit al is updated in such a direction that the lower limit of theadaptive allowable range increases, by the increasing unit change ΔINCin each control cycle to a limit provided by the lower-limit fifthpredetermined value LL in STEP10-17. At the same time, the adaptiveupper limit ah is updated in such a direction that the upper limit ofthe adaptive allowable range decreases, by the decreasing unit changeΔDEC in each control cycle to a limit provided by the upper-limit firstpredetermined value STABH.

The manner in which the adaptive allowable range thus changes with timeand also the manner in which the demand differential air-fuel ratio uslchanges with time are illustrated in FIG. 21. As shown in FIG. 21, whenthe demand differential air-fuel ratio usl lies in the adaptiveallowable range (al<usl<ah), both the adaptive upper and lower limitsah, al are updated in such a direction that the adaptive allowable rangedecreases, for thereby progressively reducing the upper and lower limitsof the adaptive allowable range.

If the demand differential air-fuel ratio usl deviates from the adaptiveallowable range beyond either one of the adaptive upper and lower limitsah, al (usl>ah or usl<al), one of the adaptive upper and lower limitsah, al beyond which the demand differential air-fuel ratio usl deviatesis updated in such a direction that the adaptive allowable rangeincreases, and hence progressively increases. At the same time, theother of the adaptive upper and lower limits ah, al is updated in such adirection that the adaptive allowable range decreases, and henceprogressively decreases.

Except when the SLD control status is in the high-level unstable stateor in the low-level unstable state, if the internal combustion engine 1is in the state immediately after the supply of fuel is cut off, orimmediately after it starts, or immediately after the end of the leanoperation mode, then the allowable range for limiting the demanddifferential air-fuel ratio usl is set to the allowable range afterFC/start/operation with lean mixture, i.e., the allowable range whoseupper limit is widest, whose upper limit AHF or AHFI is set to the fixedupper-limit fifth predetermined value AFCH. Thus, the upper limit AHF orAHFI is inhibited from having a value beyond the upper-limit fifthpredetermined value AFCH in the direction to reduce the allowable rangein STEP10-1-6.

In the allowable range after FC/start/operation with lean mixture, ifthe internal combustion engine 1 is in the idling state, then the lowerlimit ALFI is set to the fixed lower-limit third predetermined value LIwhich is the lower limit of the allowable range for idling. If theinternal combustion engine 1 is in an operating state other than theidling state, then the lower limit ALFI of the allowable range afterFC/start/operation with lean mixture is set to the variable adaptivelower limit al. The lower limit ALF (=al) is updated in each controlcycle exactly in the same manner as the adaptive lower limit al of theadaptive allowable range depending on how the demand differentialair-fuel ratio usl deviates from the allowable range afterFC/start/operation with lean mixture, except when the demanddifferential air-fuel ratio usl deviates from the allowable range afterFC/start/operation with lean mixture beyond its upper limit inSTEP10-15, STEP10-17, STEP10-21.

The manner in which the allowable range after FC, of the allowable rangeafter FC/start/operation with lean mixture, established as the allowablerange changes with time and also the manner in which the demanddifferential air-fuel ratio usl changes with time are shown in FIG. 22.Likewise, the manner in which the allowable range after start changeswith time and the manner in which the allowable range after operationwith lean mixture changes with time are shown in FIGS. 23 and 24,respectively.

As shown in FIG. 22, when the supply of fuel to the internal combustionengine 1 starts being cut off, the allowable range after FC is set up asthe allowable range for limiting the demand differential air-fuel ratiousl until a predetermined time elapses after the supply of fuel is cutoff, and the upper limit AHF thereof is fixedly maintained at theupper-limit fifth predetermined value AFCH. Therefore, the upper limitof the allowable range (the allowable range after FC) is made wide. Inthe illustrated example, the internal combustion engine 1 is in anoperating state other than the idling state, and the lower limit ALF ofthe allowable range after FC is set to the adaptive lower limit al.Basically, the lower limit ALF of the adaptive lower limit al issequentially updated in the same manner as with the adaptive allowablerange. However, if the demand differential air-fuel ratio usl exceedsthe upper limit AHF (=AFCH) of the allowable range after FC when theinternal combustion engine 1 is in the state immediately after thesupply of fuel is cut off (including the state in which the supply offuel is being cut off), the lower limit ALF (=al) of the adaptive lowerlimit al is maintained at a constant value (not updated) according tothe processing of STEP10-21.

In FIG. 22, the allowable range before the supply of fuel starts beingcut off and after elapse of a predetermined time after the supply offuel is cut off is set to the adaptive allowable range (normal operatingstate). In this embodiment, since the state in which the supply of fuelis being cut off is included in the state immediately after the supplyof fuel is cut off, the allowable range from the start of thecutting-off of the supply of fuel is set to the allowable range after FCand its upper limit AHF is set to the upper-limit fifth predeterminedvalue AFCH. However, while the supply of fuel is being cut off, theair-fuel ratio is not manipulated according to the target air-fuel ratioKCMD generated by the target air-fuel ratio generator 28. Therefore, theallowable range while the supply of fuel is being cut off may be anyarbitrary allowable range, and may not necessarily be the allowablerange after FC. Alternatively, while the supply of fuel is being cutoff, the process of limiting the demand differential air-fuel ratio uslmay not be carried out.

The allowable range after start is set up as follows: As shown in FIG.23, when the processing of the target air-fuel ratio generator 28 startsafter the internal combustion engine 1 starts to operate (the processingof the target air-fuel ratio generator 28 starts basically when theactivation of both the O₂ sensor 6 and the LAF sensor 5 is confirmed,see FIG. 9), the demand differential air-fuel ratio usl starts beingcalculated. The allowable range for limiting the demand differentialair-fuel ratio usl is set to the allowable range after start. The upperlimit AHFI thereof (in the illustrated example, the operating state ofthe internal combustion engine 1 after it starts is the idling state) isfixedly maintained at the upper-limit fifth predetermined value AFCH(whose upper limit is wide) until a predetermined time elapses after theinternal combustion engine 1 starts. Since the internal combustionengine 1 is in the idling state at this time, the lower limit ALFI ofthe allowable range after start is set to the lower-limit thirdpredetermined value LI.

If the internal combustion engine 1 after it starts is kept in theidling state, then the allowable range after elapse of the predeterminedtime after the start of the internal combustion engine 1 is basicallyset to the allowable range for idling (AHFI=HI, ALFI=LI) as shown.

The allowable range after operation with lean mixture is set up asfollows: As shown in FIG. 24, when the operation mode of the internalcombustion engine 1 changes from the lean operation mode to the normaloperation mode, the processing operation of the target air-fuel ratiogenerator 28 and the manipulation of the air-fuel ratio according to thetarget air-fuel ratio KCMD generated by the target air-fuel ratiogenerator 28 are resumed. Until a predetermined time elapses after theprocessing operation of the target air-fuel ratio generator 28 and themanipulation of the air-fuel ratio according to the target air-fuelratio KCMD generated by the target air-fuel ratio generator 28 areresumed (immediately after the end of the lean operation mode), theallowable range after operation with lean mixture is set up as theallowable range for limiting the demand differential air-fuel ratio usl,and its upper limit AHF is fixedly maintained at the upper-limit fifthpredetermined value AFCH. Therefore, the upper limit of the allowablerange (the allowable range after operation with lean mixture) is madewide. In the illustrated example, the operating state of the internalcombustion engine 1 after the lean operation mode is a normal operatingstate other than the idling state, and the lower limit ALF of theallowable range after operation with lean mixture is set to the adaptivelower limit al and sequentially updated in the same manner as with theadaptive allowable range. However, if the demand differential air-fuelratio usl exceeds the upper limit AHF (=AFCH) of the allowable rangeafter operation with lean mixture, the lower limit ALF (=al) of theallowable range after operation with lean mixture is kept constantaccording to the processing in STEP10-21.

In the lean operation mode, inasmuch as the processing operation of thetarget air-fuel ratio generator 28 is not carried out, the demanddifferential air-fuel ratio usl and the upper and lower limits AHF, ALFof the allowable range are kept at values immediately prior to the startof the lean operation mode. In the present embodiment, the period inwhich the allowable range after operation with lean mixture is set up asthe allowable range for limiting the demand differential air-fuel ratiousl extends until the predetermined time elapses after the end of thelean operation mode. However, the allowable range after operation withlean mixture may be set up until the estimated differential output VO2bar sequentially determined by the estimator 26 after the end of thelean operation mode, for example, i.e., the estimated value of theoutput of the O₂ sensor 6 determined by the estimated differentialoutput VO2 bar becomes substantially equal to the air-fuel ratioreference value FLAF/BASE.

If the SLD control status is not in the low-level unstable state or thehigh-level unstable state, or the internal combustion engine 1 is not inthe state immediately after the supply of fuel is cut off, orimmediately after it starts, or immediately after the end of the leanoperation mode, or the vehicle with the internal combustion engine 1 isin the state immediately after it starts (at this time, the internalcombustion engine 1 is not idling), then the allowable range forlimiting the demand differential air-fuel ratio usl is set to theallowable range after being driven under load (whose lower limit isnarrow) which has the lower limit ALF set to the lower-limit secondpredetermined value VSTL, and the lower limit ALF is inhibited fromhaving a value beyond the lower-limit second predetermined value VSTL inthe direction to increase the allowable range in STEP10-1-21.

In the allowable range after being driven under load, the upper limitAHF is set to the variable adaptive upper limit ah in STEP10-1-19. Theupper limit AHF (=ah) is updated in each control cycle exactly in thesame manner as the adaptive upper limit ah of the adaptive allowablerange depending on how the demand differential air-fuel ratio usldeviates from the allowable range after being driven under load, exceptwhen the demand differential air-fuel ratio usl deviates from theallowable range after being driven under load beyond its lower limit inSTEP10-15, STEP10-18, STEP10-20.

The manner in which the allowable range after being driven under loadestablished as the allowable range changes with time and also the mannerin which the demand differential air-fuel ratio usl changes with timeare shown in FIG. 25.

As shown in FIG. 25, when the vehicle starts running from the idlingstate of the internal combustion engine 1 (the internal combustionengine 1 starts to drive its load), the allowable range is set to theallowable range after being driven under load and has its lower limitALF fixedly maintained at the lower-limit second predetermined valueVSTL until a predetermined time elapses after the vehicle starts running(immediately after the vehicle starts running). Therefore, the lowerlimit of the allowable range (the allowable range after being drivenunder load) is made relatively narrow. In this case, the upper limit AHFof the allowable range after being driven under load is set to theadaptive upper limit ah and sequentially updated in the same.

While the internal combustion engine 1 is idling before the vehiclestarts running, the allowable range is normally set to the allowablerange for idling (AHFI=HI, ALFI=LI). In FIG. 25, the allowable rangeafter elapse of the predetermined time after the vehicle starts runningis.

The details of the limiting process carried out in STEP10 have beendescribed above.

Referring back to FIG. 8, the adder 31 in the target air-fuel ratiogenerator 28 adds the air-fuel ratio reference value FLAF/BASE (morespecifically, the air-fuel ratio reference value FLAF/BASE determined bythe reference value setting unit 11 in STEP12, described later on, inthe preceding control cycle) to the command differential air-determininga target air-fuel ratio KCMD in the present control cycle in STEP11.

The target air-fuel ratio KCMD thus determined is stored in atime-series fashion in a memory (not shown) in each control cycle. Whenthe engine-side control unit 7 b uses the target air-fuel ratio KCMDdetermined by the target air-fuel ratio generator 28 of the exhaust-sidecontrol unit 7 a (see STEPf in FIG. 6), the latest target air-fuel ratioKCMD stored in the time-series fashion is selected.

Then, the target air-fuel ratio generator 28 carries out a process ofsetting (updating) the air-fuel ratio reference value FLAF/BASE with thereference value setting unit 11 in STEP12, after which the processing inthe present control cycle is finished.

In the present embodiment, the air-fuel ratio reference value FLAF/BASEis defined as the sum of a predetermined fixed component flaf/base(hereinafter referred to as a “reference value fixed componentflaf/base”) and a variable component flaf/adp (hereinafter referred toas a “reference value variable component flaf/adp”), i.e.,FLAF/BASE=flaf/base+flaf/adp. For variably setting up the air-fuel ratioreference value FLAF/BASE, the value of the reference value variablecomponent flaf/adp is adjusted. In the present embodiment, the referencevalue fixed component flaf/base is regarded as a “stoichiometricair-fuel ratio”.

The processing of STEP12 is carried out according to a flowchart shownin FIG. 26.

The reference value setting unit 11 determines whether the differentialoutput VO2 of the O₂ sensor 6 is of value within a predetermined rangenear “0” (a range whose lower and upper limits are set respectively tofixed values ADL (<0), ADH (>0), hereinafter referred to as a“convergence determining range”) or not in STEP12-1. Stated otherwise,the reference value setting unit 11 determines whether the outputVO2/OUT of the O₂ sensor 6 is converged substantially to its targetvalue VO2/TARGET or not. In the present embodiment, the absolute valuesof the lower and upper limits ADL, ADH of the convergence determiningrange are identical to each other (|ADL|=|ADH|).

If the differential output VO2 falls in the convergence determiningrange (ADL<VO2<ADH) and hence the output VO2/OUT of the O₂ sensor 6 isconverged substantially to the target value VO2/TARGET, then thereference value setting unit 11 compares the value of the stabilitydetermining basic parameter Pstb (see STEP9-1 shown in FIG. 16)determined by the limiter 30 in STEP9 (the process of determining thestability of the SLD control status) with a predetermined value δ (>0)in STEP12-2, for thereby determining the stability of the SLD controlstatus as in STEP9-4 shown in FIG. 16.

The predetermined value δ to be compared with the stability determiningbasic parameter Pstb in STEP12-2 is smaller than the predetermined valueε used in STEP9-4 shown in FIG. 16, thus making stricter the conditionto determine that the SLD control status is stable.

If Pstb<δ, determining that the SLD control status is stable, then thereference value setting unit 11 adjust the reference value variablecomponent flaf/adp depending on the adaptive control law input uadp (theadaptive control law component of the demand differential air-fuel ratiousl) determined by the sliding mode controller 27 in STEP8-3 shown inFIG. 15 in STEP12-3 through STEP12-7.

More specifically, the reference value setting unit 11 compares thevalue of the adaptive control law input uadp with a predetermined rangenear “0” (a range whose lower and upper limits are set respectively topredetermined (fixed) values NRL (<0), NRH (>0), hereinafter referred toas a “reference value adjusting dead zone”) or not in STEP12-3,STEP12-5. In the present embodiment, the absolute values of the lowerand upper limits NRL, NRH of the reference value adjusting dead zone areidentical to each other (|NRL|=|NRH|).

If the adaptive control law input uadp is smaller than the lower limitNRL of the reference value adjusting dead zone (uadp<NRL), then thereference value setting unit 11 subtracts a predetermined (constant)change Δflaf (>0, hereinafter referred to as a “reference value unitchange Δflaf”) from a present value flaf/adp(k−1) (a value determined inthe preceding control cycle) of the reference value variable componentflaf/adp for thereby determining a new reference value variablecomponent flaf/adp(k) in STEP12-4. That is, the reference value variablecomponent flaf/adp is reduced by the reference value unit change Δflaf.

If the adaptive control law input uadp is greater than the upper limitNRH of the reference value adjusting dead zone (uadp>NRH), then thereference value setting unit 11 adds the reference value unit changeΔflaf to the present value flaf/adp(k−1) of the reference value variablecomponent flaf/adp for thereby determining a new reference valuevariable component flaf/adp(k) in STEP12-6. That is, the reference valuevariable component flaf/adp is increased by the reference value unitchange Δflaf.

If the adaptive control law input uadp falls in the reference valueadjusting dead zone (NRL≦uadp≦NRH), then the value of the referencevalue variable component flaf/adp is not changed, but kept at thepresent value flaf/adp(k−1) in STEP12-7.

Then, the reference value setting unit 11 adds the value of thereference value variable component flaf/adp(k) determined in either oneof STEP12-4, STEP12-6, STEP12-7 to the reference value fixed componentflaf/base for thereby determining the air-fuel ratio reference valueFLAF/BASE to be used to determine the target air-fuel ratio KCMD inSTEP11 in the next control cycle in STEP12-8. Thereafter, controlreturns to the main routine shown in FIG. 8.

If the output VO2/OUT of the O₂ sensor 6 is not converged to its targetvalue VO2/TARGET (VO2≦ADL or VO2≧ADH) in STEP12-1, or if the SLD controlstatus is unstable (Pstb≧δ) in STEP12-2, then the reference valuevariable component flaf/adp is not changed, but STEP12-7 is executed tohold the value of the reference value variable component flaf/adp at thepresent value flaf/adp(k−1). Then, STEP12-8 is executed to determine theair-fuel ratio reference value FLAF/BASE, after which control returns tothe main routine shown in FIG. 8.

The reference value variable component flaf/adp which is varieddepending on the adaptive control law input uadp is stored in anonvolatile memory (not shown), e.g., an EEPROM, so that it will not belost when the internal combustion engine 1 is stopped in operation andthe exhaust-side control unit 7 a is turned off. The stored value of thereference value variable component flaf/adp will be used as an initialvalue of the reference value variable component flaf/adp when theinternal combustion engine 1 is operated next time. The initial value ofthe internal combustion engine 1 when the internal combustion engine 1is operated for the first time is “0”.

The manner in which the adaptive control law in-put uadp varies inSTEP12 and the manner in which the reference value variable componentflaf/adp and the air-fuel ratio reference value FLAF/BASE vary dependingon the adaptive control law input uadp are shown respectively in upperand lower stages in FIG. 27.

As shown in FIG. 27, when the adaptive control law input uadp falls inthe reference value adjusting dead zone (periods T1, T3, T5), thereference value variable component flaf/adp and the air-fuel ratioreference value FLAF/BASE are not varied, but kept constant. When theadaptive control law input uadp is greater than the upper limit NRH ofthe reference value adjusting dead zone (a period T2 in FIG. 27), thereference value variable component flaf/adp and the air-fuel ratioreference value FLAF/BASE are increased by the reference value unitchange Δflaf in each control cycle. When the adaptive control law inputuadp is smaller than the lower limit NRL of the reference valueadjusting dead zone (a period T4 in FIG. 27), the reference valuevariable component flaf/adp and the air-fuel ratio reference valueFLAF/BASE are reduced by the reference value unit change Δflaf in eachcontrol cycle. In this manner, the adaptive control law input uadp isfinally converted to such a value that the adaptive control law inputuadp falls within the reference value adjusting dead zone.

Details of the operation of the plant control system according to thepresent embodiment have been described above.

The operation of the plant control system is summarized as follows:Basically, the manipulated variable generator 29 (the sliding modecontroller 27, the estimator 26, and the identifier 25) of the targetair-fuel ratio generator 28 sequentially generates the demanddifferential air-fuel ratio usl as an input to be given to the objectexhaust system E for converging the output signal VO2/OUT from the O₂sensor 6 disposed downstream of the catalytic converter 3 to the targetvalue VO2/TARGET, i.e., the difference between the air-fuel ratio of theinternal combustion engine 1 and the air-fuel ratio reference valueFLAF/BASE, required to converge the output signal VO2/OUT to the targetvalue VO2/TARGET. The air-fuel ratio reference value FLAF/BASE is addedto the command differential air-fuel ratio kcmd (basically kcmd=usl)produced by limiting the demand differential air-fuel ratio usl forthereby sequentially determining the target air-fuel ratio KCMD. Theengine-side control unit 7 b adjusts the fuel injection quantity of theinternal combustion engine 1 in order to converge the output of the LAFsensor 5 (detected air-fuel ratio) to the target air-fuel ratio KCMD forthereby feedback-controlling the air-fuel ratio of the internalcombustion engine 1 at the target air-fuel ratio KCMD. In this manner,the output signal VO2/OUT from the O₂ sensor 6 is converged to thetarget value VO2/TARGET, making the catalytic converter 3 capable ofperforming an optimum exhaust gas purifying capability regardless ofaging thereof.

In the present embodiment, the sliding mode controller 27 performs theadaptive sliding mode control process which is highly stable against theeffect of disturbances to calculate the demand differential air-fuelratio usl. In calculating the demand differential air-fuel ratio usl,the estimator 26 sequentially determines the estimated differentialoutput VO2 bar which is the estimated value of the differential outputVO2 of the O₂ sensor 6 after the total dead time d which is the sum ofthe dead time d1 of the object exhaust system E including the catalyticconverter 3 and the dead time t2 of the air-fuel ratio manipulatingsystem (comprising the internal combustion engine 1 and the engine-sidecontrol unit 7 b). The adaptive sliding mode control process constructedusing the estimated differential output VO2 bar, more specifically, theadaptive sliding mode control process constructed for converging theestimated differential output VO2 bar to “0”, is performed by thesliding mode controller 27 to determine the demand differential air-fuelratio usl.

In this fashion, it is possible to generate the demand differentialair-fuel ratio usl suitable for converging the output signal VO2/OUT ofthe O₂ sensor 6 to the target value VO2/TARGET while compensating forthe effect of the dead times d1, d2 of the object exhaust system E andthe air-fuel ratio manipulating system.

The gain coefficients a1, a2, b1 as parameters of the exhaust systemmodel (the model for expressing the behavior of the object exhaustsystem E) required for performing the adaptive sliding mode controlprocess with the sliding mode controller 27 and the process ofcalculating the estimated differential output VO2 bar with the estimator26 are sequentially identified in a real-time basis by the identifier25. The sliding mode controller 27 and estimator 26 determine the demanddifferential air-fuel ratio usl and the estimated differential outputVO2 bar using the identified gain coefficients a1 hat, a2 hat, b1 hatwhich are the identified values of the gain coefficients a1, a2, b1.

Therefore, the identified gain coefficients a1 hat, a2 hat, b1 hat havetheir values highly reliable as depending on the behavior of the objectexhaust system E, thus increasing the reliability of the demanddifferential air-fuel ratio usl and the estimated differential outputVO2 bar.

In the engine-side control unit 7 b for manipulating the air-fuel ratioof the internal combustion engine 1, primarily the adaptive controller18 as a recursive-type controller capable of appropriately compensatingfor behavioral changes of the internal combustion engine 1 controls theair-fuel ratio of the internal combustion engine 1 at the targetair-fuel ratio KCMD. Therefore, the air-fuel ratio of the internalcombustion engine 1 can accurately be controlled at the target air-fuelratio KCMD, and hence the output signal VO2/OUT of the O₂ sensor 6 canstably and accurately be converged to the target value VO2/TARGET.

In the present embodiment, the demand differential air-fuel ratio uslgenerated by the manipulated variable generator 29 for converging theoutput signal VO2/OUT of the O₂ sensor 6 to the target value VO2/TARGETis limited to produce the command differential air-fuel ratio kcmdlimited in the allowable range, by which the target air-fuel ratio KCMDis finally determined. Therefore, even when the demand differentialair-fuel ratio usl suffers a large variation such as a spike, the targetair-fuel ratio KCMD and hence the actual air-fuel ratio of the internalcombustion engine 1 are prevented from varying excessively, so that theinternal combustion engine 1 can operate stably.

In the present embodiment, the adaptive allowable range is normally setup as the allowable range for limiting the demand differential air-fuelratio usl. The adaptive upper and lower limits ah, al of the adaptiveallowable range are sequentially variably updated depending on how thedemand differential air-fuel ratio usl deviates from the allowablerange, specifically, depending on the magnitude of the demanddifferential air-fuel ratio usl with respect to the upper and lowerlimits (see FIG. 21).

Consequently, the adaptive allowable range can be made to match exactlythe range of values of the demand differential air-fuel ratio usl as aninput to be given to the object exhaust system E for converging theoutput signal VO2/OUT from the O₂ sensor 6 to the target valueVO2/TARGET.

For example, in a situation in which the range of values of the demanddifferential air-fuel ratio usl that are generated steadily isrelatively narrow, i.e., in a situation in which the demand differentialair-fuel ratio usl falls steadily in the adaptive allowable range, theadaptive allowable range can be narrowed to match the narrow range.Conversely, in a situation in which the range of values of the demanddifferential air-fuel ratio usl that are generated steadily isrelatively wide, i.e., in a situation in which the demand differentialair-fuel ratio usl frequently falls out of the adaptive allowable range,the adaptive allowable range can be widened to match the wide range. Ina situation in which the range of values of the demand differentialair-fuel ratio usl is shifted toward the upper limit (positive) of theadaptive allowable range, i.e., in a situation in which the demanddifferential air-fuel ratio usl frequently deviates from the adaptiveallowable range beyond its upper limit, the adaptive allowable range canbe shifted toward the positive limit to match the shifted range.

Thus, the demand differential air-fuel ratio usl suitable for convergingthe output signal VO2/OUT of the O₂ sensor 6 to the target valueVO2/TARGET tends to fall in the allowable range (adaptive allowablerange), resulting in increased opportunities to determine the targetair-fuel ratio KCMD using the demand differential air-fuel ratio usl asthe command differential air-fuel ratio kcmd (KCMD=usl+FLAF/BASE). Sincethe demand differential air-fuel ratio usl is generated according to theadaptive sliding mode control process, the control process forconverging the output signal VO2/OUT of the O₂ sensor 6 to the targetvalue VO2/TARGET can stably be performed.

In a situation in which the demand differential air-fuel ratio usltemporarily varies greatly into a spike due to the effect of adisturbance and deviates from the allowable range (adaptive allowablerange), the command differential air-fuel ratio kcmd is limited to avalue in the allowable range for thereby avoiding large variations ofthe target air-fuel ratio KCMD to allow the internal combustion engine 1to operate stably. In this case, because the adaptive allowable range isupdated after limiting the demand differential air-fuel ratio usl, thedetermination of the target air-fuel ratio KCMD using the aboveinappropriate target air-fuel ratio KCMD directly and the resultingmanipulation of the air-fuel ratio of the internal combustion engine 1can reliably be eliminated.

For updating the adaptive allowable range, the increasing unit changeΔINC which defines a change in one control cycle of the adaptive upperlimit ah or the adaptive lower limit al in a direction to increase theupper limit or the lower limit is greater than the decreasing unitchange ΔDEC which defines a change in one control cycle of the adaptiveupper limit ah or the adaptive lower limit al in a direction to decreasethe upper limit or the lower limit. Therefore, in a situation in whichthe demand differential air-fuel ratio usl suitable for converging theoutput signal VO2/OUT of the O₂ sensor 6 to the target value VO2/TARGETdeviates from the adaptive allowable range, the adaptive allowable rangecan quickly be changed to an allowable range in which the demanddifferential air-fuel ratio usl falls.

The adaptive upper limit ah of the adaptive allowable range is limitedto a range between the upper-limit first predetermined value STABH andthe upper-limit fourth predetermined value HH, and the adaptive lowerlimit al of the adaptive allowable range is limited to a range betweenthe lower-limit first predetermined value STABL and the lower-limitfifth predetermined value LL (see STEP10-1-11 through STEP10-1-18 shownin FIG. 20). Therefore, the target air-fuel ratio KCMD is prevented frombecoming an excessively lean or rich value which is not suitable forsmoothly operating the internal combustion engine 1.

In the present embodiment, the allowable range for limiting the demanddifferential air-fuel ratio usl can be set to the variable adaptiveallowable range, and also can be set to any of corresponding allowableranges if the SLD control status is unstable or the internal combustionengine 1 is in a certain operating state, e.g., in a state immediatelyafter the supply of fuel is cut off or in an idling state, as described.As a result, the following advantages are offered.

If the SLD control status is unstable, the output signal VO2/OUT of theO₂ sensor 6 tends to be unstable and suffer variations. However,according to the present embodiment, the relatively narrow allowablerange for unstable low level or the allowable range for unstable highlevel is established as the allowable range for limiting the demanddifferential air-fuel ratio usl. Therefore, variations of the commanddifferential air-fuel ratio kcmd and hence the target air-fuel ratioKCMD are suppressed, resulting in stability of the output signal VO2/OUTof the O₂ sensor 6.

The allowable range established if the SLD control status is unstable isnarrower when it is unstable at high level than when it is unstable atlow level (see FIG. 18). That is, the more unstable the SLD controlstatus, the narrower the allowable range. If the SLD control status isunstable at high level, therefore, variations of the commanddifferential air-fuel ratio kcmd and hence the target air-fuel ratioKCMD can be suppressed as positively as possible, so that the stabilityof the output signal VO2/OUT of the O₂ sensor 6 can reliably beachieved. If the SLD control status is unstable at low level, while theoutput signal VO2/OUT of the O₂ sensor 6 is rendered stable, the abilityof the output signal VO2/OUT to the target value VO2/TARGET is achievedto some extent.

In this embodiment, for the predetermined time (:TMSTB) after theair-fuel ratio of the internal combustion engine 1 starts beingmanipulated according to the target air-fuel ratio KCMD generated by thetarget air-fuel ratio generator 28, i.e., after the control process ofconverging the output signal VO2/OUT of the O₂ sensor 6 to the targetvalue VO2/TARGET starts, the stability of the SLD control status is notdetermined, i.e., the SLD control status is regarded as being stable(see STEPd-11 shown in FIG. 7 and STEP9-2, STEP9-3 shown in FIG. 16), sothat the allowable range for unstable low level and the allowable rangefor unstable high level are prevented from being set up as the allowablerange for limiting the demand differential air-fuel ratio usl.Specifically, immediately after the control process of converging theoutput signal VO2/OUT of the O₂ sensor 6 starts, the output signalVO2/OUT is not converged to the target value VO2/TARGET. In order toaccelerate the convergence of the output signal VO2/OUT to the targetvalue VO2/TARGET, limiting the command differential air-fuel ratio kcmdto the upper or lower limit of the allowable range is avoided as much aspossible, and the frequency at which the demand differential air-fuelratio usl becomes the command differential air-fuel ratio kcmd isincreased. Thus, the output signal VO2/OUT of the O₂ sensor 6 canquickly be brought closely to the target value VO2/TARGET.

Immediately after the supply of fuel to the internal combustion engine 1is cut off, as shown in FIG. 22, the allowable range for limiting thedemand differential air-fuel ratio usl is set to the allowable rangeafter FC with its upper limit being wide. Specifically, since a largeamount of oxygen is stored in the catalytic converter 3 while the supplyof fuel to the internal combustion engine 1 is being cut off, the outputsignal VO2/OUT of the O₂ sensor 6 is spaced away from the target valueVO2/TARGET toward a leaner value of the air-fuel ratio (the outputsignal VO2/OUT of the O₂ sensor 6 becomes smaller, see FIG. 2).Therefore, immediately after the supply of fuel to the internalcombustion engine 1 is cut off, the demand differential air-fuel ratiousl for converging the output signal VO2/OUT of the O₂ sensor 6 to thetarget value VO2/TARGET becomes large in a direction to make theair-fuel ratio richer (in the present embodiment, toward the upper limitof the allowable range in a positive direction of the demanddifferential air-fuel ratio usl), as shown in FIG. 22. Therefore, theupper limit of the allowable range after FC is set to the upper-limitfifth predetermined value AFCH, making the allowable range wider at theupper limit. It is thus possible for the command differential air-fuelratio kcmd immediately after the supply of fuel to the internalcombustion engine 1 is cut off to have a large value at the upper limitof the allowable range according to the demand differential air-fuelratio usl, allowing the output signal VO2/OUT of the O₂ sensor 6 toconverge quickly to the target value VO2/TARGET.

In the state immediately after the start of the internal combustionengine 1, as shown in FIG. 23, the allowable range for limiting thedemand differential air-fuel ratio usl is set to the allowable rangeafter start, with its upper limit made wide in the same manner asimmediately after the supply of fuel is cut off. Specifically, sinceoxygen delivered to the catalytic converter 3 upon cranking of theinternal combustion engine 1 to start same is stored in the catalyticconverter 3, immediately after the internal combustion engine 1 starts,the output signal VO2/OUT of the O₂ sensor 6 tends to be spaced awayfrom the target value VO2/TARGET toward a leaner value of the air-fuelratio. In the present embodiment, therefore, the upper limit(corresponding to the leaner value of the air-fuel ratio) of theallowable range after start is made wide in the same manner asimmediately after the supply of fuel is cut off. Therefore, the outputsignal VO2/OUT of the O₂ sensor 6 can be converged quickly to the targetvalue VO2/TAPGET.

In the state immediately after the end of the lean operation mode of theinternal combustion engine 1 (immediately after the lean operation modechanges to the normal operation mode), as shown in FIG. 24, theallowable range for limiting the demand differential air-fuel ratio uslis set to the allowable range after operation with lean mixture, withits upper limit made wide in the same manner as immediately after thesupply of fuel is cut off. Specifically, since the air-fuel ratio of theinternal combustion engine 1 is manipulated toward a leaner value duringthe lean operation mode of the internal combustion engine 1, the outputsignal VO2/OUT of the O₂ sensor 6 is spaced away from the target valueVO2/TARGET toward a leaner value of the air-fuel ratio. In the presentembodiment, therefore, the upper limit of the allowable range afteroperation with lean mixture is made wide in the same manner asimmediately after the supply of fuel is cut off. Therefore, it ispossible for the command differential air-fuel ratio kcmd immediatelyafter the end of the lean operation mode to have a large value at theupper limit of the allowable range according to the demand differentialair-fuel ratio usl. Consequently, the output signal VO2/OUT of the O₂sensor 6 can be converged quickly to the target value VO2/TARGET, andthe catalytic converter 3 can quickly achieve an appropriate exhaust gaspurifying capability.

At this time, inasmuch as the command differential air-fuel ratio kcmdhas a large value at the upper limit of the allowable range(corresponding to a richer value of the air-fuel ratio), the actualair-fuel ratio of the internal combustion engine 1 quickly becomes arich air-fuel ratio after the end of the lean operation mode. Thus, NOxabsorbed by the NOx absorber (not shown) contained in the catalyticconverters 3, 4 during the lean operation mode of the internalcombustion engine 1 can quickly be reduced. In a next cycle of the leanoperation mode, therefore, NOx in the exhaust gases can sufficiently beabsorbed by the NOx absorber contained in the catalytic converters 3, 4for exhaust gas purification.

In the state immediately after the start of the vehicle, i.e., in thestate immediately after the internal combustion engine 1 starts drivingits load, as shown in FIG. 25, the allowable range for limiting thedemand differential air-fuel ratio usl is set to the allowable rangeafter being driven under load, with its upper limit made narrow.Specifically, immediately after the internal combustion engine 1 startsdriving its load, the air-fuel ratio of the internal combustion engine 1tends to change to a leaner value. If such a situation occurs when theair-fuel ratio is changed in a direction toward a leaner value by thedemand differential air-fuel ratio usl (in the present embodiment, in anegative direction of the demand differential air-fuel ratio usl towardthe lower limit of the allowable range), then the output signal VO2/OUTof the O₂ sensor 6 is likely to change excessively to a leaner air-fuelratio with respect to the target value VO2/TARGET. According to thepresent embodiment, the lower limit of the allowable range after beingdriven under load is set to the lower-limit second predetermined valueVSTL, thus making the allowable range narrower at the lower limit. Themagnitude (absolute value) of the value of the command differentialair-fuel ratio kcmd which can be taken at the lower limit of theallowable range immediately after the start of the vehicle, i.e.,immediately after the internal combustion engine 1 starts driving itsload, is limited to a relatively small value, thus making the outputsignal VO2/OUT of the O₂ sensor 6 stable.

In the present embodiment, the demand differential air-fuel ratio usl isdefined as a value produced by subtracting the air-fuel ratio referencevalue FLAF/BASE from the air-fuel ratio to be given to the objectexhaust system E for converging the output signal VO2/OUT of the O₂sensor 6 to the target value VO2/TARGET. Therefore, the upper limit(positive) of the allowable range corresponds to a richer air-fuel ratioand the lower limit (negative) of the allowable range corresponds to aleaner air-fuel ratio. However, the demand differential air-fuel ratiousl may be defined as a value produced by subtracting the air-fuel ratioto be given to the object exhaust system E for converging the outputsignal VO2/OUT of the O₂ sensor 6 to the target value VO2/TARGET, fromthe air-fuel ratio reference value FLAF/BASE. In this case, since thesign of the demand differential air-fuel ratio usl is opposite to thesign in the embodiment, the upper limit (positive) of the allowablerange corresponds to a leaner air-fuel ratio and the lower limit(negative) of the allowable range corresponds to a richer air-fuelratio.

While the internal combustion engine 1 is idling, the allowable rangefor limiting the demand differential air-fuel ratio usl is set to theallowable range for idling with both upper and lower limits thereofbeing made relatively narrow. Specifically, in the idling state of theinternal combustion engine 1, if the air-fuel ratio of the internalcombustion engine 1 is changed largely, then the stability of the idlingstate tends to be impaired. According to the present embodiment,therefore, the narrow allowable range for idling is established as theallowable range for limiting the demand differential air-fuel ratio usl.In this manner, the output signal VO2/OUT of the O₂ sensor 6 can beconverged to the target value VO2/TARGET while keeping the idling stateof the internal combustion engine 1 stable.

According to the present embodiment, furthermore, the air-fuel ratioreference value FLAF/BASE is variably established, as described above,depending on the adaptive control law input uadp which is a componentbased on the adaptive control law (adaptive algorithm) of the demanddifferential air-fuel ratio usl generated by the sliding mode controller27. In this manner, the air-fuel ratio reference value FLAF/BASE can beof a central value in a range of values of the air-fuel ratio(=usl+FLAF/BASE) that is the sum of the air-fuel ratio reference valueFLAF/BASE and the demand differential air-fuel ratio usl, i.e., theair-fuel ratio required for converging the output signal VO2/OUT of theO₂ sensor 6 to the target value VO2/TARGET, which air-fuel ratio isbasically equal to the target air-fuel ratio KCMD. As a result, valuesof the demand differential air-fuel ratio usl which are sequentiallygenerated by the sliding mode controller 27 can be balanced betweenpositive and negative values, and the adaptive allowable range that ismade variable depending on the demand differential air-fuel ratio uslcan be balanced between its upper limit (positive) and its lower limit(negative).

Specifically, with the output signal VO2/OUT of the O₂ sensor 6 steadilyconverged to the target value VO2/TARGET, as is apparent from theequations (24)-(26), the equivalent control law input ueq and thereaching control law input urch, which are components other than theadaptive control law input uadp of the demand differential air-fuelratio usl, become “0”, and usl=uadp. The adaptive control law input uadpis thus significant as a central value in the range of values of thedemand differential air-fuel ratio usl with the output signal VO2/OUT ofthe O₂ sensor 6 steadily converged to the target value VO2/TARGET. Thesum of the adaptive control law input uadp and the air-fuel ratioreference value FLAF/BASE is significant as a central value of theair-fuel ratio required for converging the output signal VO2/OUT of theO₂ sensor 6 to the target value VO2/TARGET, i.e., the target air-fuelratio KCMD.

By adjusting the target value VO2/TARGET such that the adaptive controllaw input uadp will be of a value close to “0”, the air-fuel ratioreference value FLAF/BASE can be of a central value of the air-fuelratio required for converging the output signal VO2/OUT of the O₂ sensor6 to the target value VO2/TARGET, i.e., the target air-fuel ratio KCMD.In this embodiment, therefore, by appropriately changing the air-fuelratio reference value FLAF/BASE depending on the value of the adaptivecontrol law input uadp, the air-fuel ratio reference value FLAF/BASE isadjusted to keep the value of the adaptive control law input uadp in thereference value adjusting dead zone. In this fashion, the air-fuel ratioreference value FLAF/BASE can be adjusted to a central value of theair-fuel ratio required for converging the output signal VO2/OUT of theO₂ sensor 6 to the target value VO2/TARGET, i.e., the target air-fuelratio KCMD. As a result, values of the demand differential air-fuelratio usl can be balanced between positive and negative values, and theadaptive allowable range that is made variable depending on the demanddifferential air-fuel ratio usl can be balanced between its upper limit(positive) and its lower limit (negative). With the adaptive allowablerange balanced between its upper limit (positive) and its lower limit(negative), the demand differential air-fuel ratio usl can be limitedappropriately in a balance at both the upper and lower limits.

In this embodiment, the air-fuel ratio reference value FLAF/BASE isadjusted (updated) only when the output signal VO2/OUT of the O₂ sensor6 is substantially converged to the target value VO2/TARGET and it isdetermined that the SLD control status is stable from the stabilitydetermining basic parameter Pstb. Thus, the air-fuel ratio referencevalue FLAF/BASE is adjusted when the value of the adaptive control lawinput uadp is stabilized, and the reliability of the air-fuel ratioreference value FLAF/BASE as a central value of the air-fuel ratiorequired for converging the output signal VO2/OUT of the O₂ sensor 6 tothe target value VO2/TARGET, i.e., the target air-fuel ratio KCMD, isincreased.

In this embodiment, in adjusting the air-fuel ratio reference valueFLAF/BASE, the value of the air-fuel ratio reference value FLAF/BASE isnot changed while the adaptive control law input uadp is present in thereference value adjusting dead zone. Thus, it is possible to avoidfrequent variations of the air-fuel ratio reference value FLAF/BASE forthereby avoiding situations in which the SLD control status is unstable.

The above adjustment of the air-fuel ratio reference value FLAF/BASEoffers the following advantages:

By changing the air-fuel ratio reference value FLAF/BASE depending onthe adaptive control law input uadp, the quick response of the controlprocess for converging the output signal VO2/OUT of the O₂ sensor 6 tothe target value VO2/TARGET can be increased. Specifically, in the casewhere the air-fuel ratio reference value FLAF/BASE is constant, e.g.,FLAF/BASE=flaf/base, then if there is a steady error between the actualair-fuel ratio of the internal combustion engine 1 and the targetair-fuel ratio KCMD, then the adaptive control law input uadp determinedby the sliding mode controller 27 finally corresponds to a learned valueof the error. If the error is relatively large, then it takes a periodof time for the adaptive control law input uadp to finally correspond toa learned value of the error. According to the present invention, sincethe air-fuel ratio reference value FLAF/BASE is changed depending on theadaptive control law input uadp, the adaptive control law input uadp canbe of a sufficiently small value close to “0”. Stated otherwise, theabove error can be absorbed by the air-fuel ratio reference valueFLAF/BASE, with the result that the quick response of the controlprocess for converging the output signal VO2/OUT of the O₂ sensor 6 tothe target value VO2/TARGET can be increased.

Furthermore, by changing the air-fuel ratio reference value FLAF/BASEdepending on the adaptive control law input uadp, the accuracy of theestimated differential output VO2 bar determined by the estimator 26 andthe identified gain coefficients a1 hat, a2 hat, b1 hat determined bythe identifier 25 can be increased. The reasons for this are as follows:The exhaust system model expressed by the equation (1) with the air-fuelratio reference value FLAF/BASE used as a reference for the input to theobject exhaust system E is a model in which the output KACT (detectedair-fuel ratio) of the LAF sensor 5 becomes the air-fuel ratio referencevalue FLAF/BASE with the output signal VO2/OUT of the O₂ sensor 6steadily converged to the target value VO2/TARGET. Therefore, theair-fuel ratio reference value FLAF/BASE should be a central value ofthe air-fuel ratio of the internal combustion engine 1 with the outputsignal VO2/OUT of the O₂ sensor 6 steadily converged to the target valueVO2/TARGET. In this embodiment, by changing the air-fuel ratio referencevalue FLAF/BASE depending on the adaptive control law input uadp, theair-fuel ratio reference value FLAF/BASE can be adjusted to a centralvalue of the air-fuel ratio required for converging the output signalVO2/OUT of the O₂ sensor 6 to the target value VO2/TARGET. As aconsequence, the behavior of the exhaust system model can be made tobetter match the actual behavior of the object exhaust system.Therefore, the accuracy of the estimated differential output VO2 bardetermined by the estimator 26 based on the exhaust system model can beincreased, and the accuracy of the identified gain coefficients a1 hat,a2 hat, b1 hat determined by the identifier 25 as identified values ofparameters of the exhaust system model can be increased. With theaccuracy of the estimated differential output VO2 bar and the identifiedgain coefficients a1 hat, a2 hat, b1 being increased, the demanddifferential air-fuel ratio usl determined by the sliding modecontroller 27 using these data can be made optimum for converging theoutput signal VO2/OUT of the O₂ sensor 6 to the target value VO2/TARGET.As a result, the accuracy of the control process for converging theoutput signal VO2/OUT of the O₂ sensor 6 to the target value VO2/TARGETcan be increased.

A plant control system according to another embodiment of the presentinvention will be described below. The embodiment basically differs fromthe above embodiment only as to the processing carried out by theestimator 26. Therefore, the same reference characters as those of theabove embodiment will be used in the description of the otherembodiment.

In the above embodiment, in order to compensate for the effect of thetotal dead time d which is the sum of the dead time d1 of the objectexhaust system E and the dead time d2 of the air-fuel ratio manipulatingsystem (comprising the internal combustion engine 1 and the engine-sidecontrol unit 7 b), an estimated value of the differential output VO2 ofthe O₂ sensor 6 (estimated differential output VO2 bar) after the totaldead time d is determined. If the dead time d2 of the air-fuel ratiomanipulating system is sufficiently small as compared with the dead timed1 of the object exhaust system E, then an estimated value VO2(k+d1) ofthe differential output VO2 of the O₂ sensor 6 after the dead time d1 ofthe object exhaust system E (hereinafter referred to as a “secondestimated differential output VO2 bar”) may be determined, and thedemand differential air-fuel ratio usl may be determined using thesecond estimated differential output VO2 bar. In this embodiment, thesecond estimated differential output VO2 bar is determined to convergethe output signal VO2/OUT of the O₂ sensor 6 to the target valueVO2/TARGET.

In this case, the estimator 26 determines the second estimateddifferential output VO2 bar as follows: Using the equation (1)representing the exhaust system model of the object exhaust system E,the second estimated differential output VO2 bar which represents anestimated value of the differential output VO2 of the O₂ sensor 6 afterthe dead time d1 of the object exhaust system E in each control cycle isexpressed, using time-series data VO2(k), VO2(k−1) of the differentialoutput VO2 of the O₂ sensor 6 and time-series data kact(k−j) (j=1, 2, .. . , d1) of past values of the differential output kact(kact=KACT−FLAF/BASE) of the LAF sensor 5, according to the followingequation (42): $\begin{matrix}{{\overset{\_}{VO2}( {k + {d1}} )} = {{{\alpha 3} \cdot {{VO2}(k)}} + {{\alpha 4} \cdot {{VO2}( {k - 1} )}} + {\sum\limits_{j = 1}^{d1}{\gamma_{j} \cdot {{kact}( {k - j} )}}}}} & (42)\end{matrix}$

where

α3=the first-row, first-column element of A^(d1),

α4=the first-row, second-column element of A^(d1),

γj=the first-row elements of A^(j−1)·B $A = \begin{bmatrix}{a1} & {a2} \\1 & 0\end{bmatrix}$ $B = \begin{bmatrix}{b1} \\0\end{bmatrix}$

In the equation (42), α3, α4 represent the first-row, first-columnelement and the first-row, second-column element of the d1th powerA^(d1) (d1: total dead time of the object exhaust system E) of thematrix A defined in the equation (12), and γj (j=1, 2, . . . , d1)represents the first-row elements of the product A^(j−1)·B of the(j−1)th power A^(j−1) (j=1, 2, . . . , d1) of the matrix A and thevector B defined in the equation (12).

The above equation (42) is a basic equation for the estimator 26 tocalculate the second estimated differential output VO2(k+d1) bar in thisembodiment. The equation (42) is similar to the equation (12) exceptthat “kcmd” in the equation (12) is replaced with “kact” and “d” in theequation (12) is replaced with “d1”. In this embodiment, the equation(42) is calculated using time-series data VO2(k), VO2(k−1) of thedifferential output VO2 of the O₂ sensor 6 and time-series datakact(k−j) (j=1, 2, . . . , d1) of past values of the differential outputkact (kact=KACT−FLAF/BASE) of the LAF sensor 5 in each control cycle, todetermine the second estimated differential output VO2(k+d1) bar of theO₂ sensor 6.

The values of the coefficients α3, α4 and γj (j=1, 2, . . . , d1) whichare necessary to calculate the second estimated differential outputVO2(k+d1) bar according to the equation (42) are calculated using theidentified gain coefficients a1 hat, a2 hat, b1 hat which are identifiedvalues of the gain coefficients a1, a2, b1. The value of the dead timed1 required in the calculation of the equation (42) is the same as inthe previous embodiment.

The other processing than the processing described above is basicallythe same as in the previous embodiment. However, the sliding modecontroller 27 determines the equivalent control input ueq, the reachingcontrol law input urch, and the adaptive control law uadp, which arecomponents of the demand differential air-fuel ratio usl according tothe equations (24), (26), (27), respectively, where “d” is replaced with“d1”. In the process of limiting combinations of the identified gaincoefficients a1 hat, a2 hat, carried out by the identifier 25, dependingon the value of the dead time d1 of the object exhaust system E, therange for limiting the combinations (which corresponds to theidentifying coefficient stable range or the identifying coefficientlimiting range shown in FIG. 12) may differ from the range in theprevious embodiment, but may be established in the same manner as in theprevious embodiment.

The plant control system according to this embodiment operates on thesame manner and offers the same advantages as the with previousembodiment with respect to the process of limiting the demanddifferential air-fuel ratio usl and the variable setting of the air-fuelratio reference value FLAF/BASE.

The plant control system where the object exhaust system E serves as aplant is not limited to the above embodiments, but may be modified asfollows:

In the above embodiments, the LAF sensor (widerange air-fuel ratiosensor) 5 is employed as the second detecting means for detecting anair-fuel ratio as an input to the object exhaust system E. However, thesecond detecting means may comprise an ordinary O₂ sensor or any ofvarious other types of sensors insofar as it can detect the air-fuelratio of an air-fuel mixture combusted by the internal combustion engine1.

In the above embodiments, an output from the object exhaust system Erepresents an oxygen concentration in the exhaust gas, and the O₂ sensor6 is employed as the first detecting means for detecting the oxygenconcentration. However, the first detecting means may comprise any ofvarious other types of sensors insofar as it can detect theconcentration of a certain component of an exhaust gas downstream of thecatalytic converter to be controlled. For example, if carbon monoxide inan exhaust gas downstream of the catalytic converter is to becontrolled, the first detecting means may comprise a CO sensor. Ifnitrogen oxide (NOx) in an exhaust gas downstream of the catalyticconverter is to be controlled, the first detecting means may comprise anNOx sensor. If hydrocarbon (HC) in an exhaust gas downstream of thecatalytic converter is to be controlled, the first detecting means maycomprise an HC sensor. When a three-way catalytic converter is employed,then even if the concentration of any of the above gas components isdetected, it may be controlled to maximize the purifying performance ofthe three-way catalytic converter. If a catalytic converter foroxidation or reduction is employed, then purifying performance of thecatalytic converter can be increased by directly detecting a gascomponent to be purified.

In the above embodiments, the gain coefficients a1, a2, b1 as parametersof the exhaust system model are identified by the identifier 25.However, the gain coefficients a1, a2, b1 may be fixed to predeterminedvalues, or may be established from a map or the like depending onoperating conditions of the internal combustion engine 1 anddeteriorated states of the catalytic converter 3.

In the above embodiments, the estimator 26 and the sliding modecontroller 27 use a common exhaust system model of the object exhaustsystem E. However, the estimator 26 and the sliding mode controller 27may use respective models. In such a case, an input to the exhaustsystem model for use in the processing of the sliding mode controller 27may not necessarily be expressed using the air-fuel ratio referencevalue FLAF/BASE.

In the above embodiments, the exhaust system model is expressed by adiscrete system (discrete-time system). However, it may be expressed acontinuous system (continuous-time system), and an algorithm for theprocessing of the estimator 26 and the sliding mode controller 27 may beconstructed based on the model of such a continuous system(continuous-time system).

In the above embodiments, the adaptive sliding mode control process isused as a feedback control process for generating the manipulatedvariable (the demand differential air-fuel ratio usl) for manipulatingthe air-fuel ratio of the internal combustion engine 1 for convergingthe output signal VO2/OUT of the O₂ sensor 6 to the target valueVO2/TARGET, using the estimated differential output VO2 bar. However,any of various other feedback control processes (preferably forgenerating a manipulated variable component corresponding to theadaptive control law input uadp) may be used.

In the above embodiments, the air-fuel ratio of the internal combustionengine 1 is feedback-controlled at the target air-fuel ratio KCMD usingthe output signal from the LAF sensor 5. However, it is possible tomanipulate the air-fuel ratio of the internal combustion engine 1 intothe target air-fuel ratio KCMD by adjusting the amount of fuel suppliedto the internal combustion engine 1 under feedforward control using amap or the like based on the target air-fuel ratio KCMD.

In the above embodiments, in adjusting the air-fuel ratio referencevalue FLAF/BASE, the value of the air-fuel ratio reference valueFLAF/BASE is not changed while the adaptive control law input uadp ispresent in the reference value adjusting dead zone. However, theair-fuel ratio reference value FLAF/BASE may be changed such that if theadaptive control law input uadp is greater than “0”, then the air-fuelratio reference value FLAF/BASE is incremented by the reference valueunit change Δflaf in each control cycle, and if the adaptive control lawinput uadp is smaller than “0”, then the air-fuel ratio reference valueFLAF/BASE is decremented by the reference value unit change Δflaf ineach control cycle. The air-fuel ratio reference value FLAF/BASE canthen be adjusted such that the adaptive control law input uadp will besubstantially “0”.

In the above embodiments, the air-fuel ratio reference value FLAF/BASEis changed depending on the adaptive control law input uadp according tothe adaptive sliding mode control process. However, even if the demanddifferential air-fuel ratio usl is generated according the normalsliding mode control process which does not include the adaptive controllaw input uadp, the air-fuel ratio reference value FLAF/BASE canvariably be established. Specifically, if the demand differentialair-fuel ratio usl is determined as the sum of the equivalent controlinput ueq and the reaching control law input urch (usl=ueq+urch)according to the normal sliding mode control process which does notemploy the adaptive control law, then the reaching control law inputurch in a state wherein the value of the stability determining basicparameter Pstb (=σ bar·Δσ bar) or the value of the rate of change of theswitching function σ bar is steadily substantially “0” corresponds tothe adaptive control law input uadp in the above embodiments. Therefore,the same advantages as in the above embodiments can be achieved bychanging the air-fuel ratio reference value FLAF/BASE depending on thereaching control law input urch in the above state in the same manner asin the above embodiments.

In the above embodiments, the control system for controlling theinternal combustion engine 1 mounted on the vehicle has been describedby way of example. However, the present invention is also applicable toan engine for actuating an object other than the vehicle, e.g., anelectric generator or the like.

In the above embodiments, the plant control system where the objectexhaust system E serves as the plant has been described by way ofexample. However, the plant control system according to the presentinvention is not limited to the above embodiments.

A plant control system according to still a second embodiment of thepresent invention will be described below with reference to FIG. 28.

As shown in FIG. 28, a plant 40 is supplied with an alkaline solution ata flow rate which can be regulated by a flow rate controller (actuator)41. The plant 40 mixes the supplied alkaline solution with an acidsolution, and stirs them into a mixed solution with a stirrer 42.

The plant control system according to the embodiment shown in FIG. 28serves to control the flow rate of the alkaline solution supplied to theplant 40 for adjusting the pH of the mixed solution (the mixture of thealkaline solution and the acid solution) discharged from the plant 40 toa desired pH, i.e., a pH corresponding to a neutral value.

The plant control system has a pH sensor 43 (first detecting means)disposed at the outlet of the plant 40 for detecting the pH of the mixedsolution discharged from the plant 40, a flow rate sensor 44 (seconddetecting means) disposed at the inlet of the plant 40 for detecting theflow rate of the alkaline solution supplied to the plant 40, and acontrol unit 45 for performing a processing operation (described lateron) based on respective outputs V1/OUT, V2/OUT of the pH sensor 43 andthe flow rate sensor 44.

The control unit 45 comprises a microcomputer or the like. The controlunit 45 comprises a subtractor 46 for calculating a difference V1(=V1/OUT−V1/TARGET, which will hereinafter be referred to as adifferential output V1 of the pH sensor 43)) between the output V1/OUTof the pH sensor 43 and a target value V1/TARGET (which corresponds to atarget pH of the mixed solution) therefor, a reference value settingunit 47 (reference value setting means) for sequentially generating areference value V2/REF for the flow rate of the alkaline solution to besupplied to the plant 40 (which reference value will also be a referencevalue for the output of the flow rate sensor 44), and a subtractor 48for calculating a difference V2 (=V2/OUT−V2/REF, which will hereinafterreferred to as a differential output V2 from the flow rate sensor 44)between the output V2/OUT of the flow rate sensor 44 and the referencevalue V2/REF. The control unit 45 also comprises a manipulated variabledetermining unit 49 for determining a difference wsl (which correspondsto the demand differential air-fuel ratio kcmd in the above embodiments,and will hereinafter be referred to as a demand differential flow ratewsl) with the reference value V2/REF for the flow rate of the alkalinesolution to be supplied to the plant 40 in order to converge the outputV1/OUT of the pH sensor 43 to the target value V1/TARGET, as amanipulated variable for manipulating the output of the flow ratecontroller 41, an adder 50 for adding the reference value V2/REF to thedemand differential flow rate wsl to determine a target flow rate V2CMDfor the alkaline solution to be supplied to the plant 40, and a feedbackcontroller 51 (actuator control means) for adjusting an operationvariable (e.g., the opening of a valve) for the flow rate controller 41under feedback control for converging the output V2/OUT (detected flowrate) of the flow rate sensor 44 to the target flow rate V2CMD.

A system which comprises the flow rate controller 41 and the feedbackcontroller 51, i.e., a system for generating the output V2/OUT of theflow rate sensor 44 from the target flow rate V2CMD is referred to as aflow rate manipulating system (which corresponds to the air-fuel ratiomanipulating system in the above embodiments).

The manipulated variable determining unit 49 has an identifier, anestimator, and a sliding mode controller (not shown) which are identicalto those of the air-fuel ratio manipulated variable determining unit 29(see FIG. 3) according to the above embodiments. The manipulatedvariable determining unit 49 employs a discrete-system model of theplant 40 where VO2, kact in the equation (1) described above arereplaced respectively with the differential outputs V1, V2, and adiscrete-system model of the flow rate manipulating system where kact,kcmd in the equation (2) are replaced respectively with the differentialoutput V2 and the demand differential flow rate wsl, and carries out thesame processing operations as those of the identifier 25, the estimator26, and the sliding mode controller 27 of the air-fuel ratio manipulatedvariable determining unit 49.

Specifically, the manipulated variable determining unit 49 calculatesidentified values (which correspond to the identified gain coefficientsa1 hat, a2 hat, b1 hat in the above embodiments) of parameters of themodel of the plant 40, using the data of the differential outputs V1,V2. The manipulated variable determining unit 49 also calculates anestimated value (which corresponds to the estimated differential outputVO2 bar in the above embodiments) of the differential output V1 of thepH sensor 43 after a total dead time which is the sum of a dead timeexisting in the plant 40 and a dead time existing in the flow ratemanipulating system, using the data of the differential outputs V1, V2and the data of the identified values of the parameters of the model ofthe plant 40. The manipulated variable determining unit 49 alsocalculates the demand differential flow rate wsl according to theadaptive sliding mode control process, using the data of thedifferential outputs V1, V2, the data of the estimated value of thedifferential output V1, and the data of the identified values of theparameters of the model of the plant 40.

A preset value of the dead time in the model of the plant 40 may bedetermined by way of experimentation so as to be a time (e.g., aconstant value) which is equal to or greater than the actual dead timeof the plant 40. A preset value of the dead time in the model of theflow rate manipulating system may be determined by way ofexperimentation so as to be a time (e.g., a constant value) which isequal to or greater than the actual dead time of the flow ratemanipulating system in view of the operating characteristics of the flowrate controller 41.

For limiting the values of parameters of the model of the plant 40 to beidentified by the identifier as with the above embodiments, conditionsfor limiting the values of the parameters or their combinations may beestablished through experimentation or simulation in view of thecontrollability of the output V1/OUT of the pH sensor 43 at the targetvalue V1/TARGET, the stability of the demand differential flow rate wsl,and the stability of operation of the flow rate controller 41 dependingthereon, in the same manner as with the above embodiments.

As with the reference value setting unit 11 in the above embodiments,the reference value setting unit 47 sequentially variably determines thereference value V2/REF depending on an adaptive control law input wadpthat is determined by a sliding mode controller in the manipulatedvariable determining unit 49 as a component based on the adaptivecontrol law of the demand differential flow rate wsl according to theadaptive sliding mode control process.

The adaptive control law input wadp can be determined by an equationwhich is the same as the right-hand side of the equation (27) using theswitching function σ bar where the VO2 bar according to the equation(25) is replaced with the estimated value of the differential output V1.

As with the general feedback controller 15 according to the aboveembodiments, the feedback controller 51 feedback-controls the operationof the flow rate controller 41 to equalize the output V2/OUT (detectedflow rate) of the flow rate sensor 44 to the target flow rate V2CMD witha PID controller, an adaptive controller, or the like (not shown).

The plant control system according to the embodiment shown in FIG. 28 iseffective to converge the output V1/OUT of the pH sensor 43, i.e., thepH of the mixed solution generated by the plant 40, to a desired pH(target value V1/TARGET) according to the adaptive sliding mode controlprocess regardless of the effect of disturbances and the dead timeexisting in the plant 40, without recognizing the pH of the alkalinesolution supplied to the plant 40, the pH of the acid solution mixedwith the alkaline solution in the plant 40, and the flow rate of theacid solution.

With the reference value V2/REF relative to the model of the plant 40being sequentially established depending on the adaptive control lawinput wadp relative to the adaptive sliding mode control process, thequick response of the control process for converging the output V1/OUTof the pH sensor 43 to the target value VI/TARGET can be increased. Atthe same time, the accuracy of the estimated value of the differentialoutput V1 and the identified values of the parameters of the model ofthe plant 40 can also be increased. As a result, the accuracy of thecontrol process for converging the output V1/OUT of the pH sensor 43 tothe target value V1/TARGET can be increased.

In this embodiment, the process of limiting the demand differential flowrate wsl is omitted. However, the target flow rate V2CMD may bedetermined by limiting the demand differential flow rate wsl to a valuein a given allowable range and adding the reference value V2/REF. Insuch a case, it is also possible to variably set up the allowable rangedepending on how the demand differential flow rate wsl deviates from theallowable range, or change the manner in which the allowable range isset up depending on the operating state of the flow rate controller 41.

The plant control system according to the present embodiment may bemodified in the same manner as the above embodiments where the objectexhaust system E serves as a plant.

Although certain preferred embodiments of the present invention havebeen shown and described in detail, it should be understood that variouschanges and modifications may be made therein without departing from thescope of the appended claims.

What is claimed is:
 1. A plant control system for controlling a plant,comprising: an actuator for generating an input to the plant; firstdetecting means for detecting an output from the plant; manipulatedvariable determining means for sequentially generating a manipulatedvariable which manipulates the input to the plant to converge the outputfrom said first detecting means to a predetermined target value;actuator control means for controlling operation of said actuator basedon said manipulated variable to manipulate the input to the plant;estimating means for sequentially generating data representing anestimated value of the output from said first detecting means after adead time of said plant according to a predetermined algorithmconstructed based on a model of a behavior of the plant as a system forgenerating the difference between the output from said first detectingmeans and said target value with a response delay and the dead time fromthe difference between the input to the plant and a predeterminedreference value, said manipulated variable determining means comprisingmeans for generating said manipulated variable using the data generatedby said estimating means; and reference value variable setting means forvariably setting said predetermined reference value depending on themanipulated variable generated by said manipulated variable determiningmeans.
 2. A plant control system according to claim 1, wherein saidplant comprises an exhaust system ranging from a position upstream of acatalytic converter disposed in an exhaust passage of an internalcombustion engine for purifying an exhaust gas from the internalcombustion engine, to a position downstream of the catalytic converter,said exhaust system including said catalytic converter, said input tothe plant comprising an air-fuel ratio of an air-fuel mixture combustedin the internal combustion engine as said actuator for generating theexhaust gas which enters said catalytic converter, said output from theplant comprising the concentration of a component of the exhaust gashaving passed through said catalytic converter.
 3. A plant controlsystem according to claim 1, further comprising second detecting meansfor detecting the input to the plant, said algorithm comprising analgorithm for generating the data representing the estimated value ofthe output from said first detecting means after said dead time, using dat a representing the difference between the output of said firstdetecting means and said target value, data representing the differencebetween an output from said second detecting means and said referencevalue, and parameters of said model of the plant which define thebehavior of the model of the plant.
 4. A plant control system forcontrolling a plant, comprising: an actuator for generating an input tothe plant; first detecting means for detecting an output from the plant;manipulated variable determining means for sequentially generating amanipulated variable which manipulates the input to the plant toconverge the output from said first detecting means to a predeterminedtarget value; actuator control means for controlling operation of saidactuator based on said manipulated variable to manipulate the input tothe plant; estimating means for sequentially generating datarepresenting an estimated value of the output from said first detectingmeans after a total dead time which is the sum of a first dead time ofsaid plant and a second dead time of an input manipulating system,according to a predetermined algorithm constructed based on a model of abehavior of the plant as a system for generating the difference betweenthe output from said first detecting means and said target value with aresponse delay and the first dead time from the difference between theinput to the plant and a predetermined reference value, and a model of abehavior of the input manipulating system as a system comprising saidactuator control means and said actuator for generating the input tosaid plant with said second dead time from said manipulated variable,said manipulated variable determining means comprising means forgenerating said manipulated variable using the data generated by saidestimating means; and reference value variably setting means forvariably setting said predetermined reference value depending on themanipulated variable generated by said manipulated variable determiningmeans.
 5. A plant control system according to claim 4, wherein saidplant comprises an exhaust system ranging from a position upstream of acatalytic converter disposed in an exhaust passage of an internalcombustion engine for purifying an exhaust gas from the internalcombustion engine, to a position downstream of the catalytic converter,said exhaust system including said catalytic converter, said input tothe plant comprising an air-fuel ratio of an air-fuel mixture combustedin the internal combustion engine as said actuator for generating theexhaust gas which enters said catalytic converter, said output from theplant comprising the concentration of a component of the exhaust gashaving passed through said catalytic converter.
 6. A plant controlsystem according to claim 4, wherein said algorithm comprises analgorithm for generating the data representing the estimated value ofthe output from said first detecting means after said total dead time,using data representing the difference between the output of said firstdetecting means and said target value, data representing saidmanipulated variable, and parameters of said model of said plant whichdefine the behavior of the model of said plant.
 7. A plant controlsystem according to claim 6, further comprising second detecting meansfor detecting the input to the plant, wherein the data representing saidmanipulated variable and used by said estimating means to generate theestimated value of the output from said first detecting means includesat least one past value of said manipulated variable prior to saidsecond dead time of said input manipulating system, said algorithmcomprising an algorithm for generating the estimated value of the outputfrom said first detecting means using data representing said past valuewith an output from said second detecting means based on the model ofsaid input manipulating system, instead of all or some of past values ofsaid manipulated variable prior to said second dead time.
 8. A plantcontrol system according to claim 3, further comprising identifyingmeans for sequentially identifying the parameters of the model of saidplant, using the data representing the difference between the output ofsaid first detecting means and said target value, and the datarepresenting the difference between the output from said seconddetecting means and said reference value.
 9. A plant control systemaccording to claim 7, further comprising identifying means forsequentially identifying the parameters of the model of said plant,using the data representing the difference between the output of saidfirst detecting means and said target value, and the data representingthe difference between the output from said second detecting means andsaid reference value.
 10. A plant control system according to any one ofclaims 1 through 9, wherein said model of the plant comprises a modelexpressing the behavior of said plant with a discrete-time system.
 11. Aplant control system according to any one of claims 1 through 9, whereinsaid manipulated variable generating means comprises means forgenerating said manipulated variable in order to converge the output ofsaid first detecting means to said target value according to a feedbackcontrol process constructed based on the model of said plant.
 12. Aplant control system according to claim 10, wherein said manipulatedvariable generating means comprises means for generating saidmanipulated variable in order to converge the output of said firstdetecting means to said target value according to a feedback controlprocess constructed based on the model of said plant.
 13. A plantcontrol system according to claim 11, wherein said manipulated variablecomprises a target value for the difference between the input to saidplant and said reference value, said actuator control means comprisingmeans for controlling operation of said actuator in order to manipulatethe input to said plant into a target input determined based on saidtarget value for the difference and said reference value.
 14. A plantcontrol system according to claim 11, wherein said feedback controlprocess comprises a process of generating said manipulated variableusing data representing the difference between the estimated value ofthe output from said first detecting means represented by the datagenerated by said estimating means and said target value, and parametersof said model of the plant which define the behavior of the model of theplant.
 15. A plant control system according to claim 11, wherein saidfeedback control process comprises a sliding mode control process.
 16. Aplant control system according to claim 15, wherein said sliding modecontrol process comprises an adaptive sliding mode control process. 17.A plant control system according to claim 16, wherein said manipulatedvariable generated by said manipulated variable generating meansaccording to said adaptive sliding mode control process includes anadaptive control law component based on an adaptive control law of saidadaptive sliding mode control process, said reference value variablesetting means comprising means for variably setting said reference valuebased on the value of the adaptive control law component of saidmanipulated variable.
 18. A plant control system according to claim 17,wherein said reference value variable setting means comprises means forvariably setting said reference value by increasing or decreasing saidreference value depending on the magnitude of the value of the adaptivecontrol law component of said manipulated variable with respect to apredetermined value or a range close to and containing saidpredetermined value.
 19. A plant control system according to claim 15,wherein said reference value variable setting means comprises means forsequentially determining whether the output from said first detectingmeans is stable or not, and holding said reference value as apredetermined value irrespective of said manipulated variable if theoutput from said first detecting means is unstable.
 20. A plant controlsystem according to claim 19, wherein said reference value variablesetting means comprises means for determining whether the output fromsaid first detecting means is stable or not based on the value of aswitching function used in said sliding mode control process.
 21. Aplant control system according to any one of claims 1 through 9, whereinsaid reference value variable setting means comprises means fordetermining whether the output from said first detecting means issubstantially converged to said target value or not, and holding saidreference value as a predetermined value irrespective of saidmanipulated variable if the output from said first detecting means isnot converged to said target value.
 22. A plant control system accordingto claim 15, wherein said reference value variable setting meanscomprises means for determining whether the output from said firstdetecting means is substantially converged to said target value or not,and holding said reference value as a predetermined value irrespectiveof said manipulated variable if the output from said first detectingmeans is not converged to said target value.